Let $a_1, a_2, a_3, \dots$ be an $A$.$P$. and $g_1, g_2, g_3, \dots$ be an increasing $G$.$P$. If $a_1 = g_1$ and $a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ 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.

Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1)$,$b_n=b_{n-1}+a_{n-1}$,$\forall n \geq 2$. If $S =\sum \limits_{n=1}^{10} \frac{b_n}{2^n}$ and $T =\sum \limits_{n=1}^8 \frac{n}{2^{n-1}}$,then $2^7(2S - T)$ is equal to $........$.

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. The greatest value of $n$ for which $a_n$ is the greatest is:

Let the sum of an infinite $G.P.$,whose first term is $a$ and the common ratio is $r$,be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $A.P.$,whose first term is $10ar$,$n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$,is equal to.

For a positive integer $n$,let $a(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2^n - 1}$. Then: