Prove the following by using the principle of mathematical induction for all n ∈ N:
1/2 + 1/4 + 1/8 + .... + 1/2ⁿ = 1 - 1/2ⁿ

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1/2 + 1/4 + 1/8 + .... + 1/2ⁿ = 1 - 1/2ⁿ

For n = 1,

P (1) : 1/2¹ = 1 - 1/2¹

1/2 = 1 - 1/2

1/2 = 1/2, which is true.

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

i.e., 1/2 + 1/4 + 1/8 + .... + 1/2^k = 1 - 1/2^k ....(1)

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

Now, we have

1/2 + 1/4 + 1/8 + .... + 1/2^{k + 1}

= 1/2 + 1/4 + 1/8 + .... + 1/2^k + 1/2^{k + 1}

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

= 1 - 1/2^k (1 - 1/2)

= 1 - 1/2^k .1/2

= 1 - 1/2^{k + 1}

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 9

Prove the following by using the principle of mathematical induction for all n ∈ N: 1/2 + 1/4 + 1/8 + .... + 1/2ⁿ = 1 - 1/2ⁿ

Summary:

We have proved that 1/2 + 1/4 + 1/8 + .... + 1/2ⁿ = 1 - 1/2ⁿ by using the principle of mathematical induction for all n ∈ N