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 $