G Tewani Solutions for Chapter: Progression and Series, Exercise 5: DPP

The absolute value of the sum of first $20$ terms of series, if $S_n=\frac{n+1}{2}\text{ and }\frac{T_{n-1}}{T_n}=\frac{1}{n^2}-1$, where $n$ is odd, given $S_n$ and $T_n$ denotes sum of first $n$ terms and $n^{th}$ term of the series.

If $S_n=\left(1^2-1+1\right)(1!)+\left(2^2-2+1\right)(2!)+\dots +\left(n^2-n\right.$ $+1\left)\right(n!)$, then $S_{50}$ is:

If $S_n=\frac{1·2}{3!}+\frac{2·2^2}{4!}+\frac{3·2^2}{5!}+\dots +$ up to $n$ terms, then sum of infinite terms is: 

There is a certain sequence of positive real numbers. Beginning from the third term, each term of the sequence is the sum of all the previous terms. The seventh term is equal to $1000$ and the first term is equal to $1$. The second term of this sequence is equal to: 

The sequence $\left\{x_1, x_2\dots x_{50}\right\}$ has such a property that for each $k$, $x_k$ is $k$ less than the sum of other $49$ numbers. The value of $96x_{20}$ is:

Let $a_0=0$ and $a_n=3a_{n-1}+1$ for $n\ge 1$. Then the remainder obtained on dividing $a_{2010}$ by $11$ is

Suppose $a_1,a_2,a_3,\dots ,a_{2012}$ are integers arranged in order on a circle. Each number is equal to the average of its two adjacent numbers. If the sum of all even indexed numbers is $3018$, What is the sum of all numbers?

The sum of the series $\frac{9}{5^2·2·1}+\frac{13}{5^3·3·2}+\frac{17}{5^4·4·3}+\dots$ upto infinity: