use mathematical induction method to prove the following Sn

use mathematical induction method to prove the following

S_n = 1 + 2 + 4 + . . . + 2ⁿ = 2⁽ⁿ⁺¹⁾ - 1

Solution

For Base case Let n = 1. Then:

LHS = 1 (only the first term will be taken into account)

RHS = 2^(0+1) - 1 = 2^(1) - 1 = 1

...and:

So (*) works for n = 1.

Assume, for n = k, that (*) holds; that is, that

1 + 2 + 2² + 2³ + 2⁴ + ... + 2^k = 2^k+1