Let P(n) ≡ 2 + 3.2 + 4.22 +…..+ (n + 1) 2n-1 = n.2n , for all n ∈ N.
Step I:
Put n = 1
L.H.S. = 2
R.H.S. = 1(21 ) = 2
∴ L.H.S. = R.H.S.
∴ P(n) is true for n = 1.
Step II:
Let us assume that P(n) is true for n = k.
∴ 2 + 3.2 + 4.22 + ….. + (k + 1) 2k-1 = k.2k …..(i)
Step III:
We have to prove that P(n) is true for n = k + 1,
i.e., to prove that
2 + 3.2 + 4.22 + .... + (k + 2)2k = (k + 1)2k+1
∴ P(n) is true for n = k + 1.
Step IV:
From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.
∴ 2 + 3.2 + 4.22 +……+ (n + 1) 2n-1 = n.2n for all n ∈ N.