Prove the following by using the principle of mathematical induction for all n ∈ N:
1.2 + 2.2² + 3.2³ + ..... + n.2ⁿ = (n - 1) 2^{ⁿ ⁺ ¹} + 2

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1.2 + 2.2² + 3.2³ + ..... + n.2ⁿ = (n - 1) 2^{n + 1} + 2

For n = 1,

P (1) : 1.2 = (1 - 1) 2^{1 + 1} + 2

2 = 0 + 2

2 = 2, which is true.

Assume that P (k) is true for some positive integer k.

i.e., 1.2 + 2.2² + 3.2³ + ..... + n.2^k = (k - 1) 2^{k + 1} + 2 ....(1)

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

Now, we have

1.2 + 2.2² + 3.2³ + ..... + (k + 1).2^{k + 1}

= [1.2 + 2.2² + 3.2³ + .... + k.2ⁿ] + (k + 1).2^{k + 1}

= (k - 1) 2^{k + 1} + 2 + (k + 1).2^{k + 1} ....[from (1)]

= [(k -1) + (k + 1)] 2^{k + 1} + 2

= 2k. 2^{(k + 1)} + 2

= k. 2^{(k + 1) + 1} + 2

= [(k + 1) - 1] .2^{(k + 1) + 1} + 2

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

Hence, from the principle of mathematical induction, the statement P (n) is true for all natural numbers i.e., n ∈ N .

---

NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 8

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.2² + 3.2³ + ..... + n.2ⁿ = (n - 1) 2^{ⁿ ⁺ ¹} + 2

Summary:

We have proved that 1.2 + 2.2² + 3.2³ + ..... + n.2ⁿ = (n - 1) 2^{ⁿ ⁺ ¹} + 2 by using the principle of mathematical induction for all n ∈ N