Using the principle of mathematical induction, prove each of the following for all n ϵ N:

1.2 + 2.2² + 3.2³ + …. + n.2ⁿ =(n – 1)2ⁿ⁺¹ + 2.

To Prove:

Let us prove this question by principle of mathematical induction (PMI)

Let P(n):

For n = 1

LHS = 1 × 2 = 2

RHS = (1 - 1) × 2^{(1 + 1)} + 2

= 0 + 2 = 2

Hence, LHS = RHS

P(n) is true for n 1

Assume P(k) is true

……(1)

We will prove that P(k + 1) is true

1×

1×

1× ……(2)

We have to prove P(k + 1) from P(k), i.e. (2) from (1)

From (1)

1×

Adding both sides,

(1× +

=

=

=

(1× =

which is the same as P(k + 1)

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for

where n is a natural number

Put k = n - 1

(1× =

Hence proved.