Question

21. By using principle of mathematical induction for every natural number.Prove that
1.2 + 2.22 +3.2+
+ n.2n= (n-1)2n+1+2​

Answer 1

Answer:

Here is Your Answer ✍️

Let P(n) be the given statement i.e. P(n):1.2+2.2

2

+3.2

3

+.........+n.2

n

=(n−1)2

n+1

+2

Putting n = 1, LHS = 1.2 = 2; RHS = 0+2 = 2

∴ P(n) is true for n = 1

Assume that P(n) is true for n=k i.e. P(k) is true i.e. 1.2+ 2.3 + 3.4 + ........... + k.2

k

Replacing k by k + 1, we get the next term =(k+1)2

k+1

Adding it to both sides

LHS=1.2+2.2

2

+3.2

3

+.........+k.2

k+1

+.........+(k+1)2

k+1

RHS=(k−1)2

k+1

+2+(k+1)2

k+1

=2

k+1

[k−1+k+1]+2=2k

2

k+1

+2=k2

k+2

+2

This proves P(n) true for n = k+1

Thus P(k+1) is true whenever P(k) is true

Hence. P(n) is true for all n ∈ N