Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: `1/2 + 1/4 + 1/8 + ... + 1/2^N = 1 - 1/2^N`

Question

Prove the following by using the principle of mathematical induction for all n ∈ N: `1/2 + 1/4 + 1/8 + ... + 1/2^n = 1 - 1/2^n`

Solution

`= 1 - 1/2^k(1/2)`

`= 1 - 1/(2^(k + 1))`

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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Principle of Mathematical Induction

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