Prove `1.4.7+2.5.8+3.6.9+.......` upto n terms `=(n)/(4)(n+1)(n+6)(n+7)`

Explanation

Let `P(n):1.4.7+2.5.8+3.6.9+......+` upto n terms
`(n)/(4)(n+1)(n+6)(n+7)`
i.e., `P(n):1.4.7+2.5.8+3.6.9+......+n(n+3)(n+6)=(n)/(4)(n+1)(n+6)(n+7)`
Step I For `n=1`,
LHS of Eq. (i) `=1.4.7=28`
RHS of Eq. (i) `=(1)/(4)(1+1)(1+6)(1+7)=(2.7.8)/(4)=28`
LHS = RHS
Therefore, P(1) is true .
Step II Let us assume that the result is true for `n=k`. Then , `P(k):1.4.7+2.5.8+3.6.9+......+k(k+3)(k+6)=(k)/(4)(k+1)(k+6)(k+7)`
Step III For `n=k+1`, we have to prove that
`P(k+1):1.4.7+2.5.8+3.6.9+.....+k(k+3)(K+6)+(k+1)(k+4)(k+7)`
`=((k+1))/(4)(k+2)(k+7)(k+8)`
LHS =1.4gt7+2.5.8+3.6.9+......+k(k+3)(k+6)+(k+1)(k+4)(k+7)`
=(k)/(4)(k+1)(k+6)(k+7)+(k+1)(k+4)(k+7)`[by assumption step ]
`=(k+1)(k+7){(k)/(4)(k+6)+(k+4)}`
`=(k+1)(k+7){(k^2+6k+4k+16)/(4)}`
`=(k+1)(k+7){(k^2+10k+16)/(4)}`
`=(k+1)(k+7){((k+2)(k+8))/(4)}`
`=((k+1))/(4)(k+2)(k+7)(k+8)=RHS`
This shows that the result is true for `n=k+1`. Hence, by the principle of mathematical induction , the result is true for all `n in N`.