Question

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

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $3$

$(1+{10}^{-4}{)}^{{10}^4}$

$=1+{}^{{10}^4}{C}_1\left({10}^{-4}\right)+{}^{{10}^4}{C}_2\left({10}^{-8}\right)....$

Neglecting higher terms, we get

$=1+\left({10}^4{.10}^{-4}\right)+\frac{{10}^4\left(9999\right)}{2}{.10}^{-8}$

$=1+1+\frac{{10}^{-4}\left(9999\right)}{2}$

$=2+\frac{0.9999}{2}$

$=2+0.4995$

Approx. $2.5$ $<3$