Q. 124.2( 13 Votes )
Prove the following using the principle of mathematical induction for all n ∈ N
1.2 + 2.22 + 3.23 + …+n.2n = (n – 1)2n + 1 + 2
Answer :
Let the given statement be P(n), as
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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