Question

Prove by principal of Mathematical Induction that, 1+3+5+……….+(2n- 1) = n 2 iv) If z = x + i y and | 2z-1|= |z-2|, prove that , x 2 + y 2 = 1

plz answer the question fast it is very urgent​

Answer 1

Answer:

sorry I don't know the exact answer

Answer 2

Answer:

Let P(n): 1 + 3 + 5 + ... + (2n - 1) = n2 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) = 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