#### Question

Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: 1.2 + 2.2^{2} + 3.2^{2} + … + *n*.2^{n} = (*n* – 1) 2^{n}^{+1} + 2

#### Solution

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

P(*n*): 1.2 + 2.2^{2} + 3.2^{2} + … + *n*.2^{n} = (*n* – 1) 2^{n}^{+1} + 2

For *n* = 1, we have

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

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

1.2 + 2.2^{2} + 3.2^{2} + … + *k.*2^{k} = (*k* – 1) 2^{k}^{ + 1} + 2 … (i)

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

Consider

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*.

Is there an error in this question or solution?

Solution 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 Concept: Principle of Mathematical Induction.