Question

The positive integer just greater than $(1+0.0001{)}^{10000}$ is

• A. 4
• B. 5
• C. 2
• D. 3

Answer 1

Answer: (D)

${\left(1+0.0001\right)}^{10000}={\left(1+\frac{1}{n}\right)}^n,n=10000$
$=1+n.\frac{1}{n}+\frac{n\left(n-1\right)}{2!}\frac{1}{n^2}+\frac{n\left(n-1\right)\left(n-2\right)}{3!}\frac{1}{n^3}+.....$
$=1+1+\frac{1}{2!}\left(1-\frac{1}{n}\right)+\frac{1}{3!}\left(1-\frac{1}{n}\right)+\left(1-\frac{2}{n}\right)+....$
$<1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+.....+\frac{1}{\left(9999\right)!}$
$=1+\frac{1}{1!}+\frac{1}{2!}+.....\infty =e<3$