Question

If n is a positive integer, then $2^n>1+n\sqrt{\left(2n-1\right)}$.

• A. True
• B. False

Answer 1

The correct option is A True

We have to prove that
$\frac{2^n-1}{2-1}>n{.2}^{\frac{(n-1)}{2}}$
But we know that $\frac{a(r^n-1)}{r-1}$ is the sum of a G.P. whose
first term is a and common ratio is r.
In the above a = 1 and r = 2
$\therefore$ We have to prove that
$1+2+2^2+2^3+\dots \dots +2^{n-1}>n{.2}^{\frac{(n-1)}{2}}$
Now
$\frac{1+2+2^2+3^3\dots \dots 2^{n-1}}{n}>({\mathrm{1.2.2}}^2{.2}^3+\dots \dots 2^{n-1}{)}^{\frac{1}{n}}$
or $>(2^{1+2+3+\dots \dots n-1}{)}^{\frac{1}{n}}\because A.M.>G.M$
or $>[2^{n-1}\frac{n}{2}{]}^{\frac{1}{n}}or>2^{\frac{(n-1)}{2}}$
$1+2+2^2+\dots \dots +n=\frac{n(n+1)}{2}$