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Question

Mathematics Question on Binomial theorem

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

A

4

B

5

C

2

D

3

Answer

3

Explanation

Solution

(1+0.0001)10000=(1+1n)n,n=10000\left(1+ 0.0001\right)^{10000} = \left(1+ \frac{1}{n}\right)^{n} , n =10000 =1+n.1n+n(n−1)2!1n2+n(n−1)(n−2)3!1n3+.....= 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+12!(1−1n)+13!(1−1n)+(1−2n)+....= 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+11!+12!+13!+.....+1(9999)!< 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ..... + \frac{1}{\left(9999\right)!} =1+11!+12!+.....∞=e<3= 1 + \frac{1}{1!} + \frac{1}{2!} + ..... \infty= e < 3