Prove the following using the principle of mathematical induction for all n ∈ N
1.2 + 2.2² + 3.2³ + …+n.2ⁿ = (n – 1)2^{n + 1} + 2

Question

Prove the following using the principle of mathematical induction for all n ∈ N 1.2 + 2.2 2 + 3.2 3 + …+n.2 n = (n – 1)2 n + 1 + 2

Philoid · Accepted Answer

Let the given statement be P(n), as

First, we check if it is true for n = 1,

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

…………..(1)

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

Solving the left hand side with n = k + 1

[From equation (1)]

Which is equal to the Right hand side for n = k + 1.We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.

Prove the following using the principle of mathematical induction for all n ∈ N1.2 + 2.22 + 3.23 + …+n.2n = (n – 1)2n + 1 + 2