Find the positive integer just greater than `(1+0. 0001)^(10000)dot`

Options

A. 3
B. 4
C. 5
D. none of these

Answer

a

Explanation

For any n `in` N, we have
`(1 + (1)/(n))^(n) = 1 + n . (1)/(n)+ (n(n -1))/(2!) ((1)/(n))^(2) + (n(n-1) (-2))/(3!) ((1)/(3))^(3) + ...`
`rArr (1 + (1)/(n))^(n) = 1 + 1+ (1)/(2!)+(1 - (1)/(n))+ (1)/(3!) (1 -(1)/(n)) (1 - (2)/(n)) + ...`
`rArr (1 + (1)/(n))^(n) = 2+(1)/(2!)+(1 - (1)/(n))+ (1)/(3!) (1 -(1)/(n)) (1 - (2)/(n)) + ...`
`rArr (1 + (1)/(n))^(n) = 2 + ` a positive real number
`rArr (1 + (1)/(n))^(n) gt 2` for all n `in` N
We know that
`underset(ntoinfty)(lim)(1 + (1)/(n))^(n)lt 3 ` for all n `in` N
`rArr 2 lt (1 + 0.0001)^(10000) lt 3`
[ Putting n = 10000]
Hence , the greatest integer just then `(1+ 0.0001)^(10000) is 3`