Prove the following by using the principle of mathematical induction for all n ∈ N:
1.3 + 2.3² + 3.3³ + ..... + n.3ⁿ = [(2n - 1)3ⁿ ⁺ ¹ + 3]/4

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1.3 + 2.3² + 3.3³ + ..... + n.3ⁿ = [(2n - 1)3^{n + 1} + 3]/4

For n = 1,

P (1) : 1 . 3 = [(2 x 1 - 1) 3^{1 + 1} + 3]/4

3 = (1.3² + 3)/4

3 = 3, which is true.

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

i.e., 1.3 + 2.3² + 3.3³ + ..... + k.3^k = [(2k - 1)3^{k + 1} + 3]/4 .....(1)

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

Now, we have

1.3 + 2.3² + 3.3³ + ..... + (k + 1) 3^{k + 1}

⇒ [1.3 + 2.3² + 3.3³ + ..... + k.3^k] + (k + 1) 3^{k + 1}

⇒ [(2k - 1)3^{k + 1} + 3]/4 + (k + 1) 3^{k + 1}....[from (1)]

⇒ [(2k - 1)3^{k + 1} + 3 + 4 (k + 1) 3^{k + 1}]/4

⇒ {3^{k + 1} [(2k - 1) + 4 (k + 1)] + 3}/4

⇒ {3^{k + 1} [2k - 1 + 4k + 4] + 3}/4

⇒ {3^{k + 1} [6k + 3] + 3}/4

⇒ {3^{k + 1}.3 [2k + 1] + 3}/4

⇒ {[2k + 2 - 1].3^{(k + 1) + 1} + 3}/4

⇒ {[2 (k + 1) - 1] 3^{(k + 1) + 1} + 3}/4

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

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

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 5

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.3 + 2.3² + 3.3³ + ..... + n.3ⁿ = [(2n - 1)3ⁿ ⁺ ¹ + 3]/4

Summary:

We have proved that 1.3 + 2.3² + 3.3³ + ..... + n.3ⁿ = [(2n - 1)3ⁿ ⁺ ¹ + 3]/4 by using the principle of mathematical induction for all n ∈ N.