Question

State the whether given statement is true or false
For a positive integer n,let $S\left(n\right)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....+\frac{1}{2^n-1}$. Then prove that $S\left(100\right)<100$.

• A. True
• B. False

Answer 1

The correct option is A True

We have,

$S\left(n\right)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....+\frac{1}{2^n-1}$

$S\left(n\right)=1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7})+(\frac{1}{8}+\frac{1}{9}+.......)+.....+\frac{1}{2^n-1}$

$S\left(n\right)=1+(\frac{1}{2}+\frac{1}{2^2-1})+(\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{2^3-1})+(\frac{1}{8}+\frac{1}{9}+.......+\frac{1}{2^4-1})+.....+\frac{1}{2^n-1}$

Since,

$(\frac{1}{2}+\frac{1}{3})<1$

$(\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7})<1$

$\frac{1}{2^n-1}<1$

So,

$S\left(n\right)=1+1+1+......+1n\left(times\right)$

$S\left(n\right)<n$

Therefore,

$S\left(100\right)<100$

Hence, this is the answer.