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 $\_\_\_\_$。

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its...

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. $b_1, b_2, \ldots, b_n$. If...

$\frac{6}{3^{26}}+\frac{10 \cdot 1}{3^{25}}+\frac{10 \cdot 2}{3^{24}}+\frac{10 \cdot 2^2}{3^{23}}+\ldots+\frac{10 \cdot 2^{24}}{3}$ is equal to :

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a>2$. If $\alpha$ is such that $a, 4, \alpha, b$ are in A.P., then the...

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

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 equal to

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 $2 \sum_{n=1}^{40}\left(\mathrm{P}_n\right)^{\frac{1}{n}}=\frac{3^\alpha-1}{3^\beta}$ and $\operatorname{gcd}(\alpha, \beta)=1$,...

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

JEE Main Mathematics Medium

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