Prove that 1.2 + 2.3+.......n(n + 1) = n(n + 1)(n + 2)/3 in mathematical induction.

Solution:

Given 1.2 + 2.3+.......n(n + 1) = n(n + 1)(n + 2)/3

We shall prove the result by principle of mathematical induction.

Checking for n = 1,

LHS: 1.2 = 2

RHS: (1/3) × 1 × 2 × 3 = 2.

Hence true for n = 1

Let us assume the result is true for n = k ie.,

1.2 + 2.3 + .....k(k + 1) = 1/3 × k × (k + 1) × (k + 2)

We shall prove the result to be true for n = k + 1.

That is, to prove 1.2 + 2.3.....+ k(k + 1) + (k + 1)(k + 2) = 1/3(k + 1)(k + 2)(k + 3)

Consider LHS:1.2 + 2.3.....+k(k + 1) + (k + 1)(k + 2)

= 1/3 × k × (k + 1) × (k + 2) + (k + 1)(k + 2)

= (k + 1)(k + 2)[1/3(k + 1)]

= (k + 1)(k + 2)(k + 3)1/3 = RHS.

Hence the result holds for n = k + 1.

Hence proof is complete by the principle of mathematical induction and therefore the result holds.

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Prove that 1.2 + 2.3+.......n(n + 1) = n(n + 1)(n + 2)/3 in mathematical induction.

Summary:

1.2 + 2.3+.......n(n + 1) = n(n + 1)(n + 2)/3 in mathematical induction is proved.