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Question

Using the principle of mathematical induction, prove that

1.2.3+2.3.4+3.4.5 +...+n(n+1)(n+2)=n(n+1)(n+2)(n+3)4, for all nN.


Solution

Let P(n) be the statement given by

P(n):1.2.3+2.3.4+...+n(n+1)(n+2)=n(n+1)(n+2)(n+3)4

Put n=1

LHS: P(1)=1.2.3=6
RHS : P(1)=1(1+1)(1+2)(1+3)4=1×2×3×44=6         

So, P(1) is true. Now, assume that, P(k) is true, for some natural number k, such that

1.2.3+2.3.4+...+k(k+1)(k+2)=k(k+1)(k+2)(k+3)4...(i)

We will now to show that P(k+1) is true, whenever P(k) is true.

On adding (k+1),(k+2)(k+3) to both sides of eq. (i), we get

1.2.3+2.3.4+...+k(k+1)(k+2)+(k+1)(k+2)(k+3)

= k(k+1)(k+2)(k+3)4+(k+1)(k+2)(k+3)                                                           [using Eq.     (i)]

= (k+1)(k+2)(k+3)(k4+1)=(k+1)(k+2)(k+3)(k+4)4=(k+1)((k+1)+1)((k+1)+2)((k+1)+3)4       

So, P(k+1) is true.

Hence, by principle of mathematical induction, P(n) is true,  nN

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