Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that: 1 + 3 + 5 + ... + (2n – 1) = n2

Question

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

1 + 3 + 5 + ... + (2n – 1) = n²

Sum

Solution

Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 + ... + (2n – 1) = n², for n ∈ N.

Note that P(1) is true

Since P(1) : 1 = 1²

Assume that P(k) is true for some k ∈ N

i.e., P(k) : 1 + 3 + 5 + ... + (2k – 1) = k²

Now, to prove that P(k + 1) is true

We have 1 + 3 + 5 + ... + (2k – 1) + (2k + 1)

= k + (2k + 1)

= k² + 2k + 1

= (k + 1)²

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

Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N.

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Principle of Mathematical Induction

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Chapter 4: Principle of Mathematical Induction - Solved Examples [Page 61]

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NCERT Exemplar Mathematics Exemplar [English] Class 11

Chapter 4 Principle of Mathematical Induction
Solved Examples | Q 1 | Page 61

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