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

Let the given statement be P(n), as

Steps involved in solving a statement by mathematical Induction are:

Step 1: Verify that P(1) is true.
Step 2: If P(1) is true then P(2) is also true.

Step 1:

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.

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