How to find [ (1+0.0001)^10000] where [.] indicates greatest integer function is 2?

Answer 1

[ (1+0.0001)^10000] ≈
[ 2.7181459268252248640376646749131 ] = 2
The greatest integer smaller than 2.718 . . . is 2.

Answer 2

Lets expand (1+x)^10000

1 + 10000x + (10000 * 9999)/2 * x^2 + (10000 * 9999 * 9998) / 6 * x^3 + ..... + x^10000

Now here, we have
x = .0001 = 10^-4
x^2 = 10^-8
x^3 = 10^-12

10000 is 10^4
(10000 * 9999)/2 is roughly (10^8)/2
(10000 * 9999 * 9998) / 6 is roughly (10^12) / 6


10000x = 10^4 * 10^-4 = 1
(10000 * 9999)/2 * x^2 is (10^8)/2 * 10^-8 = 1/2
10000 * 9999 * 9998) / 6 * x^3 is (10^12) / 6 * 10^-12 = 1/6

It's safe to say that all the other terms of this are getting very small, very quickly so we want to know | 1 + 1 + 1/2 + 1/6 | which is 2

Answer 3

= ([1 + 0.0001]^10,000) * 2
= (1.0001^10,000) * 2
= 2.71814592682436 * 2
= 5.43629185364871