Prove the following by using the principle of mathematical induction for all n ∈ N:

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Solution

Let the given statement be P(n), i.e.,

P(n): 1 + 3 + 3² + …+ 3ⁿ^–1 =

For n = 1, we have

P(1): 1 =, which is true.

Let P(k) be true for some positive integer k, i.e.,

We shall now prove that P(k + 1) is true.

Consider

1 + 3 + 3² + … + 3^k^–1 + 3⁽^k^{+1) – 1}

= (1 + 3 + 3² +… + 3^k^–1) + 3^k

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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