Prove the following by the principle of mathematical induction: `\ 1. 2+2. 3+3. 4++n(n+1)=(n(n+1)(n+2))/3`

Explanation

Given,
`P(n) i.e.,
1.2+2.3+3.4+......+n(n+1)=[(n(n+1)(n+2))/3​]`

For n=1,

`1.2=2*(1(1+1)(1+2)​)/3=(1.2.3)/3​=2`
which is true.

Let us assume P(k) be true for positive integer k

`1.2+2.3+3.4+.....+k.(k+1)=[(k(k+1)(k+2))/3​]........(i)`
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