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Sequences and Series

Get chapter-wise JEE Main & Advanced questions with solutions

Q JEE Main 2024
If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\cdots$. upto $\infty=2\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where a and b are integers with $...
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For $x \geq 0$, the least value of $K$, for which $4^{1+x}+4^{1-x}, \frac{K}{2}, 16^x+16^{-x}$ are three consecutive terms of an...
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Q JEE Main 2024
If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\cdots .+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\cdots .+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}$, then $\alpha$ is equal to-
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Q JEE Main 2024
Let a, ar, $a r^{2^2} \ldots \ldots .$. be an infinite G.P. If $\sum_{n=0}^{\infty} a^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$,...
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Q JEE Main 2024
The value of $\frac{1 \times 2^2+2 \times 3^2+\cdots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\cdots+100^2 \times 101}$ is
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Q JEE Main 2024
Let three real numbers 𝑎, 𝑏, 𝑐 be in arithmetic progression and a +1, b, c +3 be in geometric...
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Q JEE MAIN 2024
Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3,7,11, \ldots \ldots$. If $40
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Q JEE MAIN 2024
If three successive terms of a G.P. with common ratio $r(r>1)$ are the lengths of the sides of a triangle...
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Q Jee Main 2024
If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1=\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic...
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Q Jee Main 2024
If $\log _e a, \log _e b, \log _e c$ are in an A.P. and $...
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Q JEE-Main 2024
If $2 \tan ^2 \theta-5 \sec \theta=1$ has exactly 7 solutions in the interval $\left[0, \frac{n \pi}{2}\right]$, for the least...
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Q JEE MAIN 2024
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of...
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Q Jee Main 2024
Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$, terms of a non-constant A.P. be respectively the $...
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Q JEE-Main 2024
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then
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Q JEE Main 2024
Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text {th }}$ term of a GP, whose...
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Q JEE-Main 2024
The $20^{\text {th }}$ term from the end of the progression $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}$ is...
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Q JEE MAIN 2026
If $\sum_{r=1}^{2 s}\left(\frac{r}{r^4+r^2+1}\right)=\frac{p}{q}$, where $p$ and $q$ are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$。
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Q JEE MAIN 2026
In a G.P., if the product of the first three terms is 27 and the set of all possible values...
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Q JEE MAIN 2026
The common difference of the A.P: $a_1, a_2, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P....
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Q JEE MAIN 2026
If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{\sin ^2 \mathrm{C}}{\sin ^2 \mathrm{~A}}=1, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \in\left(0, \frac{\pi}{2}\right)$, then
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Q JEE MAIN 2026
$\frac{6}{3^{26}}+\frac{10 \cdot 1}{3^{25}}+\frac{10 \cdot 2}{3^{24}}+\frac{10 \cdot 2^2}{3^{23}}+\ldots+\frac{10 \cdot 2^{24}}{3}$ is equal to :
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Q JEE MAIN 2026
Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a>2$. If $\alpha$ is such that $a, 4, \alpha, b$...
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Q JEE MAIN 2026
Let $\sum_{k=1}^n a_k=\alpha n^2+\beta n$. If $a_{10}=59$ and $a_6=7 a_1$, then $\alpha+\beta$ is equal to
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Q JEE MAIN 2026
Consider an A.P.: $a_1, a_2, \ldots, a_{\mathrm{n}} ; a_1>0$. If $a_2-a_1=\frac{-3}{4}, a_{\mathrm{n}}=\frac{1}{4} a_1$, and $\sum_{\mathrm{i}=1}^{\mathrm{n}} a_{\mathrm{i}}=\frac{525}{2}$, then $\sum_{\mathrm{i}=1}^{17} a_1$ is...
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Q JEE MAIN 2026
Let $729,81,9,1, \ldots$. be a sequence and $\mathrm{P}_n$ denote the product of the first $n$ terms of this sequence. If...
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Q JEE MAIN 2026
Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $a^2, 2 b^2, c^2$ are in G.P. If $a
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Q JEE MAIN_2026_
If the sum of the first four terms of an A.P. is 6 and the sum of its first six...
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Q JEE MAIN 2026
Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1+a_2+\ldots+a_{10}=62$, then $a_1$ is equal to:
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Q JEE MAIN 2026
The positive integer $n$, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+\cdots+(x+2 n-2)(x+2 n)=\frac{8 n}{3}$ are two consecutive even integers,...
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Q JEE MAIN 2026
Let $a_1=1$ and for $n \geq 1, a_{n+1}=\frac{1}{2} a_n+\frac{n^2-2 n-1}{n^2(n+1)^2}$. Then $\left|\sum_{n=1}^{\infty}\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to
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Q JEE MAIN 2026
Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2 \cdot a_3 \cdot a_4=64$ and...
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Q _ JEE MAIN_2019_
The sum $$ \frac{3 \times 1}{1^2}+\frac{5 \times\left(1^3+2^3\right)}{1^2+2^2}+\frac{7 \times\left(1^3+2^3+3^3\right)}{1^2+2^2+3^2}+\ldots $$ upto $10^{\text {th }}$ term, is.
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Q JEE MAIN_2019_
If $a_1, a_2, a_3, \ldots \ldots .$, an are in A.P. and $a_1+a_4+a_7+ a_{16}+a_{16}=114$, then $a_1+a_6+a_{11}+a_{16}$ is equal to:
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Q JEE MAIN 2019
If $\alpha$ and $\beta$ are the roots of the equation $375 x^2-25 x-2=0$, then $...
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Q JEE MAIN 2019
If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $...
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Q JEE MAIN 2020
Let $S$ be the sum of the first 9 terms of the series : $\{x+k a\}+\left\{x^2+(k+2) a\right\}+\left\{x^3+(k+4) a\right\}+\left\{x^4+(k+6) a\right\}+\ldots$ where...
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Q JEE-MAIN 2019
Let $S_k=\frac{1+2+3+\ldots .+k}{k}$. If $\mathrm{S}_1^2+\mathrm{S}_2^2+\ldots \ldots+\mathrm{S}_{10}^2=\frac{5}{12} \mathrm{~A}$, then A is equal to
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Q JEE MAIN 2020
If the sum of first 11 terms of an A.P., $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots$ is $0\left(\mathbf{a}_1 \neq 0\right)$, then the...
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Q JEE Main 2020
The sum of the first three terms of a G.P. is S and their product is 27. Then all such...
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Q JEE MAINN 2019
If $a_1, a_2, a_3, \ldots .$. are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first 15 terms...
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Q JEE-MAIN 2019
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first...
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Q JEE Main 2020
If $|x|
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Q JEE MAIN 2019
Let $\mathrm{a}, \mathrm{b}$ and c be the $7^{\text {th }}, 11^{\text {th }}$ and $13^{\text {th }}$ terms respectively of...
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Q JEE MAIN 2020
The product $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}}$. to $\infty$ is equal to
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Q JEE MAIN 2020
If $2^{10}+2^9 \cdot 3^1+2^8 \cdot 3^2+\ldots .+2 \cdot 3^9+3^{10}=S-2^{11}$, then $S$ is equal to:
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Q JEE MAIN 2020
If $3^{2 \sin 2 \alpha-1}, 14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an A.P. for...
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Q JEE MAIN 2019
if the sum of the first 15 terms of the series $\left(\frac{3}{4}\right)^3+\left(1 \frac{1}{2}\right)^3+\left(2 \frac{1}{4}\right)^3+3^3+\left(3 \frac{3}{4}\right)^3+\ldots$. is equal to $225 k$,...
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Q JEE MAIN 2020
Let $a, b, c, d$ and $p$ be any non zero distinct real numbers such that $...
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Q JEE MAIN 2019
If $\sin ^4 \alpha+4 \cos ^4 \beta+2=4 \sqrt{2} \sin \alpha \cos \beta ; \alpha, \beta \in[0, \pi]$, then $...
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Q JEE MAIN 2020
The common difference of the A.P. $b_1, b_2, \ldots, b_m$ is 2 more than the common difference of A.P. $...
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Q JEE MAIN 2019
Let $\mathrm{a}, \mathrm{b}$ and c be the $7^{\text {th }}, 11^{\text {th }}$ and $13^{\text {th }}$ terms respectively of...
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Q JEE MAIN 2019
The sum of the following series $$ 1+6+\frac{9\left(1^2+2^2+3^2\right)}{7}+\frac{12\left(1^2+2^2+3^2+4^2\right)}{9}+\frac{15\left(1^2+2^2+\ldots \ldots+5^2\right.}{11}+\ldots . $$ up to 15 terms, is
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Q JEE MAIN 2020
The sum $\sum_{k=1}^{20}(1+2+3+\ldots . .+k)$ is $\_\_\_\_$ .
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Q JEE MAIN 2019
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its...
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Q JEE MAIN 2020
Let $f: R \rightarrow R$ be such that for all $x \in R\left(2^{1+x}+2^{1-x}\right), f(x)$ and $\left(3^x+3^{-x}\right)$ are in A.P., then...
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Q JEE MAIN 2019
Let $a_1, a_2, \ldots, a_{10}$ be a G.P. If $\frac{a_3}{a_1}=25$, then $\frac{a_9}{a_5}$ equals
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Q JEE MAIN 2019
Let $\mathrm{a}, \mathrm{b}$ and c be in G.P. with common ratio r , where $\mathrm{a} \neq 0$ and $0
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Q JEE MAIN 2019
The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\ldots \ldots$ $$ +\frac{1^3+2^3+3^3+\ldots \ldots+15^3}{1+2+3+\ldots \ldots+15}-\frac{1}{2}(1+2+3 \ldots .+15) $$ is equal to :
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Q JEE-MAIN 2020
If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum...
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Q JEE MAIN 2020
The minimum value of $2^{\sin x}+2^{\cos x}$ is :
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Q JEE MAIN 2020
Let $a_1, a_2, \ldots a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2+\ldots+$ an. If $...
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Q JEE MAIN 2020
The number of terms common to the two A.P.’s 3, 7, 11,.....,407 and 2, 9, 16,.....,709 is______.
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Q JEE MAIN 2020
If $1+\left(1-2^2 \cdot 1\right)+\left(1-4^2 \cdot 3\right)+\left(1-6^2 \cdot 5\right)+\ldots \ldots+ \left(1-20^2 \cdot 19\right)=\alpha-220 \beta$, then an ordered pair $(\alpha, \beta)$ is...
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Q JEE Main 2019
Which primitive unit cell has unequal edge lengths (a ≠ b ≠ c) and all axial angles different from 90°?
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Q JEE MAIN 2020
$...
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Q JEE MAIN 2020
Let $\alpha$ and $\beta$ be the roots of $x^2-3 x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6 x+q=0$....
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Q JEE MAIN 2020
https://competishun.com/let-s-be-the-set-of-all-integer-solutions-x-y-z-of-the-system-of-equationsbr-beginaligned-x-2-y5-z0-2-x4-yz0-7-x14-y9-z0-endaligned-such-that-15-leq-x2y2z2-leq-150-then-the-number-of-elements-in-the-s/
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Q JEE MAIN 2019
Let $\alpha$ and $\beta$ the roots of the quadratic equation $x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0
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Q JEE MAIN 2020
The sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal toThe sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal to
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Q JEE MAIN 2019
If 19 th term of a non-zero A.P. is zero, then its $\left(49^{\text {th }}\right.$ term) $(29$ th term $)$...
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Q JEE MAIN 2020
Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of a G.P. of positive terms. If $\sum_{n=1}^{100} a_{2 n+1}=200$ and $...
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Q JEE MAIN 2020
If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10}$, then...
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Q JEE MAIN 2019
Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $...
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Q JEE-MAIN 2020
If the sum of the series $20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .$. upto $n$th term is 488 and the $n$th term...
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Q JEE MAIN 2019
If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a...
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Q JEE MAIN 2019
The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 + ......
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Q JEE Main 2021
If $S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\ldots$, then 160 S is equal to $\_\_\_\_$ .
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Q JEE MAIN 2020
Five number are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers...
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Q JEE MAIN 2020
The greatest positive integer k , for which $49^{\mathrm{k}}+1$ is a factor of the sum $49^{125}+49^{124}+\ldots \ldots+49^2+49+1$, is
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Q JEE Main 2021
Let $a_1, a_2, a 3, \ldots$ be an A.P. If $\frac{a_1+a_2+\ldots+a_{10}}{a_1+a_2+\ldots+a_p}=\frac{100}{p^2}, p \neq 10$, then $\frac{a_{11}}{a_{10}}$ is equal to :
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Q JEE MAIN 2021
Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the...
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Q JEE MAIN 2021
The sum of 10 terms of the series $\frac{3}{1^2 \times 2^2}+\frac{5}{2^2 \times 3^2}+\frac{7}{3^2 \times 4^2}+\ldots$ is :
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Q JEE MAIN 2021
If for $x, y \in R, x>0$, $y=\log _{10} x+\log _{10} x^{1 / 3}+\log _{10} x^{1 / 9}+\ldots \ldots$ upto...
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Q JEE Main 2021
If the sum of an infinite GP a, ar, $a r^2, a r^3, \ldots$ is 15 and the sum of...
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Q JEE Main 2021
The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\ldots .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is
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Q JEE Main 2019
If a, b and c be three distinct real numbers in G.P. and a + b + c = xb,...
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Q JEE MAIN 2021
Let $a_1, a_2 \ldots \ldots a_{10}$ be an AP with common difference -3 and $b_1, b_2 \ldots \ldots, b_{10}$ be...
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Q JEE Main 2019
Let $a_1, a_2 \ldots . a_{30}$ be an A.P., $S=\sum_{i=1}^{30}$ ai and $T=\sum_{i=1}^{15} a_{(2 i-1)}$. If $a_5=27$ and $S-2 T=75$,...
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Q JEE MAIN 2021
Let $\mathrm{a}_1, \mathrm{a}_2 \ldots \ldots, \mathrm{a}_{21}$ be an AP such that $\sum_{\mathrm{n}=1}^{20} \frac{1}{\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+1}}=\frac{4}{9}$. If the sum of this AP...
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Q JEE MAIN 2021
Let $S_n=1 \cdot(n-1)+2 \cdot(n-2)+3 \cdot(n-3)+\ldots \cdot+(n-1) \cdot n \geq 4$. The sum $\sum_{n=4}^{\infty}\left(\frac{2 S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to :
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Q JEE MAIN_2021
If the value of $\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\ldots . \text { upto } \infty\right)^{\log _{1008}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^2}+\ldots \text { uptoss }\right)}$ is $l$, then $l^2$...
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Q JEE MAIN 2019
The sum $\sum_{k=1}^{20} k \frac{1}{2^k}$ is equal to :
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Q JEE MAIN_2021_
Let $S_n$ be the sum of the first $n$ terms of ah arithmetic progression. If $S_{3 n}=3 S_{2 n}$, then...
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Q JEE MAIN 2019
If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^2+2 b x+c=0$ and $...
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Q JEE MAIN 2021
If the arithmetic mean and geometric mean of the $p^{\text {th }}$ and $q^{\text {th }}$ terms of the sequence...
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Q JEE MAIN 2021
If $\log _3 2, \log _3\left(2^x-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to $\_\_\_\_$...
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Q JEE MAIN 2021
The sum of the series $\sum_{n=1}^{\infty} \frac{n^2+6 n+10}{(2 n+1)!}$ is equal to:
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Q JEE MAIN 2021
If $[x]$ be the greatest integer less than or equal to $x$, then $\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]$ is equal to:
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Q _JEE MAIN_2021
Let $S_n$ denote the sum of first $n$-terms of an arithmetic progression. If $S_{10}=530, S_5=140$, then $S_{20}-S_6$ is equal to
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Q JEE MAIN 2021
Let $\left\{a_n\right\}_{n=1}^{\infty}$ be a sequence such that $a_1=1, a_2=1$ and $a_{n+2}=2 a_{n+1}+a_n$ for all $n \geq 1$. Then the value...
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Q JEE MAIN 2021
The minimum value of $f(x)=a^{a x}+a^{1-a x}$, where $a, x \in R$ and $a>0$, is equal to :
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Q JEE MAIN 2021
For $k \in N$, let $\frac{1}{\alpha(\alpha+1)(\alpha+2) \ldots .(\alpha+20)}=\sum_{K=0}^{20} \frac{A_k}{\alpha+k}$, where $\alpha>0$. Then the value of $100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^2$ is equal to $\_\_\_\_$...
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Q JEE MAIN 2021
Let $S_1$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S_2$ be the sum of...
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Q JEE MAIN 2021
If sum of the first 21 terms of the series $...
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Q JEE MAIN 2021
The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals...
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Q JEE MAIN 2021
Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be in arithmetic progression. Let the centroid of the triangle with vertices $(\mathrm{a}, \mathrm{c}),(2, \mathrm{~b})$ and...
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Q JEE MAIN 2021
If the curve $y=a x^2+b x+c, x \in R$, passes through the point $(1,2)$ and the tangent line to this...
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Q JEE MAIN 2021
The missing value in the following figure is
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Q JEE MAIN 2021
$\frac{1}{3^2-1}+\frac{1}{5^2-1}+\frac{1}{7^2-1}+\ldots \ldots+\frac{1}{(201)^2-1}$ is equal to
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Q JEE MAIN 2021
If $\alpha, \beta$ are natural numbers such that $100^\alpha-199 \beta=$ (100) $(100)+(99)(101)+(98)(102)+\ldots . .+(1)$ (199), then the slope of the...
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Q JEE-MAIN 2021
Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26,...
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Q JEE MAIN 2021
The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\ldots$ is equal to
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Q JEE MAIN 2021
In a increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of...
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Q JEE MAIN 2021
If $0
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Q JEE MAIN_2022
Consider two G.Ps. $2,2^2, 2^3$, ann and $4,4^2, 4^3$ of 60 and $n$ terms respectively. If the geometric mean of...
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Q JEE MAIN 2022
Let $a_1=b_1=1, a_n=a_{n-1}+2$ and $b_n=a_n+b_{n-1}$ for every natural number $n \geq 2$. Then $\sum_{n=1}^{15} a_n \cdot b_n$ is equal to$\_\_\_\_$...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
In a solid A B. A atoms are in ccp arrangement and $B$ atoms occupy all the octahedral sites. If...
JEE Main Physics Easy
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Q JEE MAIN 2022
Let for $\mathrm{n}=1,2, \ldots, 50, \mathrm{~S}_{\mathrm{n}}$ be the sum of the infinite geometric progression whose first term is $\mathrm{n}^2$ and...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
If $a_1(>0), a_2, a_3, a_4, a_5$ are in a G.P., $a_2+a_4=2 a_3+1$ and $3 a_2+a_3=2 a_4$, then $a_2+a_4+2 a_6$ is...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
The sum of all the elements of the set $\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}$ is $\_\_\_\_$
JEE Main Mathematics Hard
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Q JEE MAIN 2022
The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\frac{51}{6^5}+\frac{70}{6^6}+\ldots .$. is equal to :
JEE Main Mathematics Medium
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Q JEE MAIN 2022
If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the...
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Q JEE MAIN 2022
If $a_1, a_2, a_3 \ldots$. and $b_1, b_2, b_3 \ldots$. are A.P. and $a_1,=2, a_{10}=3, a_1 b_1=1=a_{10} b_{10}$ then $...
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Q JEE MAIN 2022
Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\ldots$. then $4 S$ is equal to
JEE Main Mathematics Medium
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Q JEE MAIN 2022
If $A=\sum_{n=1}^{\infty} \frac{1}{\left(3+(-1)^n\right)^n}$ and $B \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\frac{A}{B}$ is equal to :
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Q JEE MAIN 2022
Let $x, y>0$. If $x^3 y^2=2^{15}$, then the least value of $3 x+2 y$ is
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Q JEE MAIN 2022
Let $\left\{a_n\right\}_{n-0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2 a_{n+1}-a_n+1$ for all $n \geq 0$. Then, $\sum_{n-2}^{\infty} \frac{a_n}{7^n}$ is...
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Q JEE MAIN 2022
The sum $1+2 \cdot 3+3 \cdot 3^2+\ldots \ldots+10 \cdot 3^9$ is equal to
JEE Main Mathematics Easy
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Q JEE MAIN 2022
Let $\mathrm{A}=\left\{1, \mathrm{a}_1, \mathrm{a}_2 \ldots . . \mathrm{a}_{18}, 77\right\}$ be a set of integers with $1
JEE Main Mathematics Hard
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Q JEE MAIN 2022
If the sum of the first ten terms of the series $\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\ldots .$. is $\frac{m}{n}$, where $m$ and $n$ are...
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Q JEE MAIN 2022
Let $\mathrm{A}=\sum_{i=1}^{10} \sum_{j=1}^{10} \min \{\mathrm{i}, \mathrm{j}\}$ and $\mathrm{B}=\sum_{i=1}^{10} \sum_{j=1}^{10} \max \{\mathrm{i}, \beta\}$. Then $\mathrm{A}+\mathrm{B}$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Medium
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Q JEE MAIN 2022
Let $A_1, A_2, A_3, \ldots \ldots$ be an increasing geometric progression of positive real numbers. If $A_1 A_3 A_5 A_7=\frac{1}{1296}$...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
The greatest integer less than or equal to the sum of first 100 terms of the sequence $...
JEE Main Mathematics Hard
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Q JEE MAIN 2022
If $x=\sum_{n=0}^{\infty} a^n, y=\sum_{n=0}^{\infty} b^n, z=\sum_{n=0}^{\infty} c^n$, where $a, b, c$ are in A.P. and $|a|
JEE Main Mathematics Medium
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Q JEE MAIN 2022
If $\left\{a_i\right\}_{i=1}^n$ where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum_{i=1}^n a_i=192$,...
JEE Main Mathematics Easy
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Q JEE MAIN 2023
Suppose $a_1, a_2, 2, a_3, a_4$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression...
JEE Main Mathematics Easy
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Q JEE MAIN 2023
Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Medium
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Q JEE MAIN 2023
If $\operatorname{gcd}(m, n)=1$ and $1^2-2^2+3^2-4^2+\ldots \ldots+(2021)^2-(2022)^2+(2023)^2=1012 m^2 n$, then $m^2-n^2$ is equal to
JEE Main Mathematics Hard
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Q JEE MAIN 2023
If $\mathrm{S}_{\mathrm{n}}=4+11+21+34+50+$ $\_\_\_\_$ to n terms, then $\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_9\right)$ is equal to
JEE Main Mathematics Easy
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Q JEE MAIN 2023
Let $a_1, a_2, a_{3, \ldots . .}$ be a G.P. of increasing positive numbers. Let the sum of its 6th...
JEE Main Mathematics Easy
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Q JEE-Main 08-04-2023
Let an be the nth term of the series 5 + 8 + 14 + 23 + 35 + 50...
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Q JEE MAIN 2025
If the sum of the first 10 terms of the series $...
JEE Main Mathematics Hard
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Q JEE MAIN 2025
$ \begin{aligned} & \text { If } \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\ldots \infty=\frac{\pi^4}{90} \\ & \frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\ldots \infty=\alpha, \\ & \frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\ldots \infty=\beta \end{aligned} $ then...
JEE Main Mathematics Medium
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Q JEE MAIN_2025
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the...
JEE Main Physics Easy
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Q JEE MAIN_2025
Let $a_n$ be the $n^n$ term of an A.P. If $S_n=a_1+a_2+a_3+\ldots+a_n=700, a_0=7$ and $S_7=7$, then $a_n$ is equal to:
JEE Main Physics Easy
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Q JEE MAIN 2025
The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
If the sum of the first 20 terms of the series $$ \frac{4 \cdot 1}{4+3 \cdot 1^2+1^4}+\frac{4 \cdot 2}{4+3 \cdot...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the...
JEE Main Physics Medium
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Q JEE MAIN 2023
The sum to 20 terms of the series $2.2^2-3^2+2.4^2-5^2+2.6^2-\ldots \ldots$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Medium
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Q JEE MAIN 2023
Let $s 1, s 2, s s_{3000-u s 10}$ respectively be the sum to 12 terms of 10 A.P.s whose...
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Q JEE MAIN 2023
If the sum of the series
JEE Main Mathematics Medium
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Q JEE MAIN 2023
Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive...
JEE Main Mathematics Easy
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Q JEE-Main 2023
The sum of the first 20 terms of the series 5 +11 + 19 + 29 + 41 + ......
JEE Main Mathematics Easy
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Q JEE MAIN 2023
Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a G.P. Let their mean and variance be...
JEE Main Mathematics Hard
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Q JEE MAIN 2023
Let $S=109+\frac{108}{5}+\frac{107}{5^2}+\ldots \ldots \ldots+\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $\left(16 S-(25)^{-54}\right)$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Easy
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Q JEE MAIN 2023
If $a_n$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}, n=1,2,3 \ldots \ldots$, then $a$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Hard
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Q JEE MAIN 2023
Let $S_k=\frac{1+2+\ldots \ldots+K}{K}$ and $\sum_{j=1}^n S_j^2=\frac{n}{A}\left(B n^2+C n+D\right)$, where $A, B, C, D \in N$ and $A$ has least value....
JEE Main Mathematics Hard
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Q JEE MAIN 2023
The sum of all those terms, of the arithmetic progression 3, 8, 13,…… 373, which are not divisible by 3,...
JEE Main Mathematics Easy
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Q JEE MAIN 2023
Let $\left\langle a_n\right\rangle$ be a sequence such that $a_1+a_2+\ldots .+a_n=\frac{n^2+3 n}{(n+1)(n+2)}$. If $28 \sum_{k=1}^{10} \frac{1}{a_k}=p_1 p_2 p_3 \ldots p_m$, where...
JEE Main Mathematics Medium
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Q JEE MAIN 2023
Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum...
JEE Main Mathematics Medium
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Q JEE-Main 2023
Let a1, a2, a3 ...... an be n positive consecutive terms of an arithmetic progression. If d > 0 is...
JEE Main Mathematics Easy
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Q JEE-Main
The sum of the first 20 terms of the series 5 +11 + 19 + 29 + 41 + ......
JEE Main Mathematics Easy
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Q JEE-Main 2024
Let the first term of a series be $\mathrm{T}_1=6$ and its $\mathrm{r}^{\text {th }}$ term $T_r=3 T_{r-1}+6^r, r=2,3, \ldots, n$....
JEE Main Mathematics Hard
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Q JEE MAIN 2024
Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let $...
JEE Main Mathematics Medium
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Q JEE MAIN
If the sum of series $\frac{1}{1 \cdot(1+d)}+\frac{1}{(1+d)(1+2 d)}+\cdots \ldots+\frac{1}{(1+9 d)(1+10 d)}$ is equal to 5 , then 50 d is...
JEE Main Mathematics Easy
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Q JEE MAIN 2024
Let the positive integers be written in the form : If the k^"th " row contains exactly k numbers for...
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Q JEE MAIN 2024
If 1/(√1+√2)+1/(√2+√3)+⋯+1/(√99+√100)=m and 1/(1⋅2)+1/(2⋅3)+⋯+1/(99⋅100)=n, then the point (m,n) lies on the line
JEE Main Mathematics Easy
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Q JEE MAIN 2024
Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $...
JEE Main Mathematics Easy
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Q JEE-Main 2024
Let A={n∈[100,700]∩N:n is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A...
JEE Main Mathematics Medium
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Q JEE MAIN 2024
Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\cdots \ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10} n^4$. If $4 \alpha-\beta=55 k+40$, then k is equal to
JEE Main Mathematics Easy
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Q JEE-Main 01.02.24_(S1)
Let 3,7,11,15,….,403 and 2,5,8,11,…,404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to
JEE Main Mathematics Easy
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Q JEE-Main 2024
Let 3,a,b,c be in A.P. and 3,a-1,b+1,c+9 be in G.P. Then, the arithmetic mean of a,b and c is :
JEE Main Mathematics Easy
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Q JEE MAIN 2024
The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\cdots$. up to 10 terms is
JEE Main Mathematics Medium
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Q JEE MAIN
let $S_a$ denote the sum of first n terms an arithmetic progression. If S_20=790 and S_10=145, then S_15- S_5 is...
JEE Main Mathematics Easy
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Q JEE MAIN 2024
For 0
JEE Main Mathematics Medium
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Q JEE MAIN
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the...
JEE Main Mathematics Easy
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Q JEE MAIN 2024
If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \infty$, then the value of $p$ is
JEE Main Mathematics Easy
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Q JEE MAIN 2024
The number of common terms in the progressions 4,9,14,19,....., up to 25th term and 3,6,9,12, up to 37th term is...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Let $a_{1}, a_{2}, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_{1}+\left(a_{5}+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}=2233$. Then $a_{1}+a_{2}+a_{3}+\ldots+a_{2024}$ is equal to $\_\_\_\_$ .
JEE Main Mathematics Easy
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Q JEE MAIN 2025
For positive integers $n$, if $4 a_{n}=\left(n^{2}+5 n+6\right)$ and $S_{n}=\sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$, then the value of $507 S_{2025}$ is:
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Q JEE MAIN 2025
Let ${x_1},{x_2},{x_3},{x_4}$ be in a geometric progression. If 2,7,9,5 are subtracted respectively from ${x_1},{x_2},{x_3},{x_4}$, then the resulting numbers are in...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
Let $A = \{ 1,\,\,6,\,\,11,\,\,16, \ldots \} $ and $B = \{ 9,16,23,30, \ldots \} $ be the sets consisting...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
$1 + 3 + {5^2} + 7 + {9^2} + .....$ upto 40 terms is equal to
JEE Main Mathematics Easy
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Q JEE MAIN 2025
The sum 1 + 3 + 11 + 25 + 45 + 71 + ……… upto 20 terms, is equal...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
Let ${a_1},{a_2},{a_3}, \ldots $ be a G.P. of increasing positive numbers. If ${a_3}{a_5} = 729$ and $...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to:
JEE Main Mathematics Easy
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Q JEE MAIN 2025
If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^{2}}(5+2 \alpha)+\frac{1}{7^{3}}(5+3 \alpha)+\ldots \ldots \ldots$, then the value of $\alpha$ is :
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Q JEE MAIN 2025
Let ${a_1},{a_2},{a_3}, \ldots $. be in an A.P. such that $...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
The roots of the quadratic equation $3 x^{2}-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
The value of $...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Let ${{\rm{T}}_{\rm{r}}}$ be the ${{\rm{r}}^{{\rm{th }}}}$ term of an A.P. If for some ${\rm{m}},{{\rm{T}}_{\rm{m}}} = \frac{1}{{25}},\;{{\rm{T}}_{25}} = \frac{1}{{20}}$ , and...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Let $\left\langle {{a_{\rm{n}}}} \right\rangle $ be a sequence such that ${a_0} = 0,{a_1} = \frac{1}{2}$ and $...
JEE Main Mathematics Medium
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Q JEE MAIN 2025
Let ${S_n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + \frac{1}{{20}} + \ldots $ upto $n$ terms. If the sum of...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
If the first term of an A.P. is 3 and the sum of its first four terms is equal to...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
If $\sum_{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}, then {\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{1}{T_r}\right)}$ is equal to :
JEE Main Mathematics Medium
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Q JEE MAIN 2025
Suppose that the number of terms in an A.P. is $2 k, k \in \mathbf{N}$. If the sum of all...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Let ${{a}_{1}},{{a}_{2}},{{a}_{3}},\ldots $ be a G.P. of increasing positive terms. If ${{a}_{1}}{{a}_{5}}=28$ and ${{a}_{2}}+{{a}_{4}}=29$, then ${{a}_{6}}$ is equal to :
JEE Main Mathematics Easy
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Q JEE-Advanced 2025
Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)>0$...
JEE Advance Mathematics Easy
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