Question

Prove each of the following by the principle of mathematical induction​

Answer 1

Let '1 + 2 + 2^(2) + ... 2^(n - 1)' be statement P(n).

Note that:

P(n) = 2^(0) + 2^(1) + 2^(2) + ... 2^(n - 1)

For n = 1 : P(1):

LHS: 2^(1 - 1) = 2^(0) = 1

RHS: 2^(1) - 1 = 2 - 1 = 1

As RHS = LHS, P(1) is true for n = 1

Let it be true for n = m, P(m) :

1 + 2 + 2^(2) + ... 2^(m - 1) = 2^(m) - 1

For n = m + 1: P(m + 1): LHS:

=> 1 + 2 + 2^(2) + ... 2^(m+1 - 1)

=> 1 + 2 + 2^(2) + ... 2^(m)

=> 1 + 2 + 2^(2) + ... + 2^(m - 1) + 2^(m)

=> 2^(m) - 1 + 2^(m)

=> 2.2^(m) - 1

=> 2^(m + 1) - 1

RHS:

=> 2^(m + 1) - 1

As both are equal, the given statement is true.