M L Aggarwal Solutions for Chapter: Sequences and Series, Exercise 6: EXERCISE 10.6

Find the sum of the following series to $n$ terms $8+88+888+\dots$

Find the sum of the following series to $n$ terms $0.5+0.55+0.555+\dots$

Sum the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\dots$ to $n$ terms.

If $S_1,S_2$ and $S$ denote respectively the sum of $n, 2n$ and infinite number of terms of a $G.P.$ then prove that  $S_1\left(S_1-S\right)=\left(S_1-S_2\right)S$.

If $y=1+x+x^2+\dots$ to  $\infty ,x<1$ numerically, then show that $x=\frac{y-1}{y}$.

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

A snail starts moving towards a point $3\mathrm{cm}$ away at a pace of $1\mathrm{cm}$ per hour. As it gets tired, it covers only half the distance compared to previous hour in each succeeding hour. In how much time will the snail reach his target?

The inventor of chessboard was a very clever man. He asked the king, a reward of one grain of wheat for the first square, $2$ grains for the second, $4$ grains for the third, and so on, doubling the amount of the grain for each subsequent square. How many grains would have to be given?