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Question
By the principle of mathematical induction, prove that, for n ≥ 1
12 + 32 + 52 + ... + (2n − 1)2 = `("n"(2"n" - 1)(2"n" + 1))/3`
Solution
Let P(n) = 12 + 32 + 52 + ... + (2n − 1)2 = `("n"(2"n" - 1)(2"n" + 1))/3`
For n = 1
P(1) = (2 × 1 − 1)2
= `(1(2 xx 1 - 1)(2 xx 1 + 1))/3`
⇒ 1 = `(1 xx 1 xx 3)/3`
∴ P(1) is true
Let P(n) be true for n = k
∴ P(k) = 12 + 32 + 52 + ... + (2k − 1)2 = `("n"(2"k" - 1)(2"k" + 1))/3` ......(i)
For n = k + 1
R.H.S = `(("k" + 1)(2"k" + 1)(2"k" + 3))/3`
L.H.S = `("k"(2"k" - 1)(2"k" + 1))/3 ("k" + 1)^2` .....(Using (i)]
= `(2"k" + 1) [("k"(2"k" - 1))/3 + (2"k" + 1)]`
= `(2"k" + 1) [(2"k"^2- "k" + 6"k" + 3)/3]`
=`((2"k" - 1)(2"k"^2 + 5"k" + 3))/3`
= `(("k" + 1)("k" + 1)(2"k" + 3))/3`
= `(("k" + 1)(2"k" + 1)(2"k" + 3))/3`
∴ P(k + 1) is true
Thus P(k) is true
⇒ P(k + 1) is true.
Hence by principle of mathematical induction,
P(k) is true for all n ∈ N.
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