Prove the following by using the principle of mathematical induction for all n ∈ N:
1 + 3 + 3² + ... + 3ⁿ ⁻ ¹ = (3ⁿ - 1)/2

Solution:

Let P (n) be the given statement.

i.e., P (n): 1 + 3 + 3² + ... + 3^{n - 1} = (3ⁿ - 1)/2

For n = 1,

P (1) : 1 = (3¹ - 1)/2 = 2/2 = 1, which is true.

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

i.e., P (k): 1 + 3 + 3² + ... + 3^{k - 1} = (3^k - 1)/2 ....(1)

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

Now, we have

1 + 3 + 3² + ... + 3^{(k + 1) - 1}

= 1+ 3 + 3² + ... + 3^k

= (1+ 3 + 3² + ... + 3^k-1) + 3^k

= (3^k-1)/2 + 3^k

= (3^k - 1 + 2 x 3^k)/2

= [(1 + 2)3^k - 1]/2

= (3 x 3^k - 1)/2

= [3^{(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 .

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 1

Prove the following by using the principle of mathematical induction for all n ∈ N: 1 + 3 + 3² + ... + 3ⁿ ⁻ ¹ = (3ⁿ - 1)/2

Summary:

We have proved 1 + 3 + 3² + ... + 3ⁿ ⁻ ¹ = (3ⁿ - 1)/2 by using the principle of mathematical induction for all n ∈ N