Question 8:

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

1.2 + 2.2² + 3.2² + … + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

Answer

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

P(n): 1.2 + 2.2² + 3.2² + … + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

For n = 1, we have

P(1): 1.2 = 2 = (1 – 1) 2¹⁺¹ + 2 = 0 + 2 = 2, which is true.

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

1.2 + 2.2² + 3.2² + … + k.2^k = (k – 1) 2^{k + 1} + 2 … (i)

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

Consider

{1.2 + 2.2² + 3.2³ + …+k.2^k} + (k + 1)∙2^k+1

=(k – 1)2^k+1 + 2 + (k + 1)2^k+1

=2^k+1{(k – 1) + (k + 1)} + 2

=2^k+1.2k + 2

={(k + 1) – 1}2^(k+1)+1 + 2

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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