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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.

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To Prove:

1 x 21 + 2 x 22 + 3 x 23 + …. + n x 2n =(n – 1)2n+1 + 2.

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

Let P(n): 1 x 21 + 2 x 22 + 3 x 23 + …. + n x 2n

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 x 21 + 2 x 22 + 3 x 23 + …. + k x 2k = (k – 1) x 2k+1 + 2.....(1)

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

1 x 21 + 2 x 22 + 3 x 23 + (k x 1) x 2k + 1 = ((k + 1) - 1) x 2(k+1) + 1 + 2

1 x 21 + 2 x 22 + 3 x 23 + (k x 1) x 2k + 1 = (k) x 2k+2 + 2

1 x 21 + 2 x 22 + 3 x 2+ k2k + (k x 1) x 2k + 1 = (k) x 2k+2 + 2.....(2)

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

From (1)

1 x 21 + 2 x 22 + 3 x 23+ k x 2k = (k – 1) x 2k+1 + 2

Adding (k + 1) x 2k + 1 both sides,

(1 x 21 + 2 x 22 + 3 x 23+ k x 2k) + (k + 1) x 2k + 1 = (k - 1) x 2k + 1 + 2 + (k + 1) x 2k + 1

= k x 2k + 1 - 2k + 1 + + k x 2k + 1 + 2k + 1

= 2k x 2k + 1 + 2

= k x 2k + 2 + 2

(1 x 21 + 2 x 22 + 3 x 23+ k x 2k) + (k + 1) x 2k + 1 = k x 2k + 2 + 2

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 x 21 + 2 x 22 + 3 x 23) + n x 2n = (n – 1) x 2n+1 + 2

Hence proved.

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