Using the principle of mathematical induction, prove each of the following for all n ϵ N:
12 + 32 + 52 + 72 + … + (2n – 1)2 =
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