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 $|a|<1,|b|<1,|c|<1 \neq 0$, then

If $\left\{a_i\right\}_{i=1}^n$ where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum_{i=1}^n a_i=192$, $\sum_{i=1}^{n / 2} a_{2 i}=120$,...

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 is 2 and the sum...

Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to $\_\_\_\_$ .

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

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

Let $a_1, a_2, a_{3, \ldots . .}$ be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2...

Let an be the nth term of the series 5 + 8 + 14 + 23 + 35 + 50 + … and . Then$...

If the sum of the first 10 terms of the series $\frac{4 \cdot 1}{1+4 \cdot 1^4}+\frac{4 \cdot 2}{1+4 \cdot 2^4}+\frac{4 \cdot 3}{1+4 \cdot 3^4}+\ldots$. is...

$ \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 $\frac{\alpha}{\beta}$ is equal to