Let $a_{1}=1$ and for $n \ge 1$, $a_{n+1} = \frac{1}{2}a_{n} + \frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$. Then $|\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})|$ is equal to ........... .

Find the $7^{\text{th}}$ term in the following sequence whose $n^{\text{th}}$ term is $a_{n} = \frac{n^{2}}{2^{n}}$.

Let $\alpha = \sum_{n=101}^{200} 2^n \sum_{k=101}^n \frac{1}{k !}$ and $b = \sum_{n=101}^{200} \frac{2^{201}-2^n}{n !}$. Then,$\frac{a}{b}$ is

Suppose the sequence $a_1, a_2, a_3, \ldots$ is an arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means between two distinct positive numbers $a$ and $b$. Then $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$ is equal to

Write the first five terms of the sequence defined by $a_{1} = -1$ and $a_{n} = \frac{a_{n-1}}{n}$ for $n \geq 2$,and obtain the corresponding series.