Question

Using method of mathematical induction, show that
1. 1+3+5+
(2n - 1) = n2, n EN.​

Answer 1

Step-by-step explanation:

Let P(n): 1 + 3 + 5 + ... + (2n - 1) = n² be the given statement

Put n = 1

Then, L.H.S = 1

R.H.S = (1)² = 1

. L.H.S = R.H.S.

 P(n) is true for n = 1

 Assume that P(n) is true for n = k.

 1 + 3 + 5 + ..... + (2k - 1) = k²

Adding 2k + 1 on both sides, we get

1 + 3 + 5 ..... + (2k - 1) + (2k + 1) = k² + (2k + 1) = (k + 1)²

 1 + 3 + 5 + ..... + (2k -1) + (2(k + 1) - 1) = (k + 1)²

 P(n) is true for n = k + 1.

 by the principle of mathematical induction P(n) is true for all natural numbers 'n'

Hence, 1 + 3 + 5 + ..... + (2n - 1) =n² for all n ϵ n