Maharashtra State BoardHSC Science (General) 11th
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Prove by method of induction, for all n ∈ N: 1.2 + 2.3 + 3.4 + ..... + n(n + 1) = n3(n+1)(n+2) - Mathematics and Statistics

Sum

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.

Concept: Principle of Mathematical Induction
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.1 | Q 6 | Page 73
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