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) :

For n = 1, we have

P(1): 1.3 = 3, 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 + 2.3² + 3.3³ + … + k3^k+ (k + 1) 3^k⁺¹

= (1.3 + 2.3² + 3.3³ + …+ k.3^k) + (k + 1) 3^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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