Sarvesh K Verma Solutions for Chapter: Sequence, Series & Progressions, Exercise 2: Introductory Exercise

Find the sum of the series $1·3^2+2·5^2+3·7^2+\dots$ to $20$ terms:

Find the sum of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots$ to $20$ terms :

The sum of an infinite $\mathrm{G}.\mathrm{P}.$ is $16$ and the sum of the squares of its terms is $153\frac{3}{5}.$ Find the fourth term of the progression:

If $\mathrm{x}=1+\mathrm{a}+{\mathrm{a}}^2+\dots$ to $\infty$ and $\mathrm{y}=1+\mathrm{b}+{\mathrm{b}}^2+\dots$ to $\infty ,$ and $\left|\mathrm{a}\right|<1,\left|\mathrm{b}\right|<1,$ then the value of : $1+\mathrm{ab}+{\mathrm{a}}^2{\mathrm{b}}^2+\dots$ to $\infty$ is :

A person is entitled to receive an annual payment which for each year is less by one tenth of what it was for the year before. If the first payment is $100,$ then find the maximum possible payment which he can receive, however long he may live :

Find the sum of the series $\frac{1}{4}-\frac{3}{16}+\frac{9}{64}-\frac{27}{256}+\dots$

The sum of first two terms of a $\mathrm{G}.\mathrm{P}.$ is $\frac{5}{3}$ and the sum to infinity of the series is $3.$ . Find the first term :

A ball is dropped from a height of $96\mathrm{ft}$ and it rebounds $\frac{2}{3}$ of the height it falls. If it continues to fall and rebound, find the total distance that the ball can travel before coming to rest.