Q. 6

# Using the princip

To Prove:

12 + 32 + 52 + 72 + … + (2n – 1)2 = Steps to prove by mathematical induction:

Let P(n) be a statement involving the natural number n such that

(i) P(1) is true

(ii) P(k + 1) is true, whenever P(k) is true

Then P(n) is true for all n ϵ N

Therefore,

Let P(n): 12 + 32 + 52 + 72 + … + (2n – 1)2 = Step 1:

P(1) = = 1

Therefore, P(1) is true

Step 2:

Let P(k) is true Then,

P(k): 12 + 32 + 52 + 72 + … + (2k – 1)2 = Now,

12 + 32 + 52 + 72 + … + (2(k + 1)–1)2 = = = = = = (Splitting the middle term)

= = P(k + 1)

Hence, P(k + 1) is true whenever P(k) is true

Hence, by the principle of mathematical induction, we have

12 + 32 + 52 + 72 + … + (2n – 1)2 = for all n ϵ N

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