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Question
Prove by method of induction, for all n ∈ N:
1.2 + 2.3 + 3.4 + ..... + n(n + 1) = `"n"/3 ("n" + 1)("n" + 2)`
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Solution
Let P(n) ≡ 1.2 + 2.3 + 3.4 + ..... + n(n + 1) = `"n"/3 ("n" + 1)("n" + 2)`, for all n ∈ N
Step 1:
For n = 1
L.H.S. = 1.2 = 2
R.H.S. = `1/3(1 + 1)(1 + 2)` = 2
∴ L.H.S. = R.H.S. for n = 1
∴ P(1) is true.
Step 2:
Let us assume that for some k ∈ N, P(k) is true,
i.e., 1.2 + 2.3 + 3.4 + ... + k(k + 1) = `"k"/3("k" + 1)("k" + 2)` ...(1)
Step 3:
To prove that P(k + 1) is true,
i.e., to prove that
1.2 + 2.3 + 3.4 + ..... + k(k + 1) +(k + 1)(k + 2) = `(("k" + 1))/3("k" + 2)("k" + 3)`
Now, L.H.S. = 1.2 + 2.3 + 3.4 + ... + k(k + 1) + (k + 1)(k + 2)
= `"k"/3("k" + 1)("k" + 2) + ("k" + 1)("k" + 2)` ...[By (1)]
= `("k" + 1)("k" + 2)("k"/3 + 1)`
= `(("k" + 1)("k" + 2)("k" + 3))/3`
= R.H.S.
∴ P(k + 1) is true.
Step 4:
From all the above steps and by the principle of mathematical induction, the result P(n) is true for all n ∈ N,
i.e., 1.2 + 2.3 + 3.4 + ..... + n(n + 1) = `"n"/3 ("n" + 1)("n" + 2)`, for all n ∈ N.
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