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Question
By the principle of Mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`
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Solution
Let P(n) = 1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`
For n = 1
P(1) = 1(1 + 1)
= `((1 + 1)(1 + 2))/3`
⇒ 2 = 2
∴ P(1) is true
Let P(n) b true for n = k
∴ P(k) = 1.2 + 2.3 + 3.4 +... + k(k + 1)
= `[("k"("k" + 1)("k" + 2))/3]` ......(i)
For n = k + 1
P(k + 1) = 1.2 + 2.3 + 3.4 .... + k(k + 1) + (k + 1)(k + 2)
= `(""("k" + 1)("k" + 2))/3 + ("k" + )("k"+ 2)` .....[Using (i)]
= `("k" + 1)("k" + )["k"/3 + 1]`
= `("k" + 1)("k" + 2)[(("k" + 3))/3]`
=`(("k" + 1)("k" + 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(n) is true for all n ∈ N
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