Question

D = { n€n : 1/2 (1+(-1)^n )}​

Answer 1

Step-by-step explanation:

Let P(n): 1 + 3 + 5 + ..... + (2n - 1) = n

2

be the given statement

Step 1: Put n = 1

Then, L.H.S = 1

R.H.S = (1)

2

= 1

∴. L.H.S = R.H.S.

⇒ P(n) istrue for n = 1

Step 2: Assume that P(n) istrue for n = k.

∴ 1 + 3 + 5 + ..... + (2k - 1) = k

2

Adding 2k + 1 on both sides, we get

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

2

+ (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) =n

2

, for all n ϵ n