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Question Number 177931 by Spillover last updated on 11/Oct/22

Prove by the principle of

induction that

1.4.7 + 2.5.8 + 3.6.9 + \dots n (n + 3) (n + 6)

= \frac{n}{4} (n + 1) (n + 6) (n + 7)

Answered by Ar Brandon last updated on 11/Oct/22

Test for k = 1, k = 2,

assume P_{k} is true for n and deduce that {it}^{'} s true for n + 1

P_{n} : \underset{k = 1}{\sum^{n}} k (k + 3) (k + 6) = \frac{n}{4} (n + 1) (n + 6) (n + 7)

P_{n + 1} : \underset{k = 1}{\sum^{n + 1}} k (k + 3) (k + 6) = P_{n} + {(n + 1)}^{th} term

= \frac{n}{4} (n + 1) (n + 6) (n + 7) + (n + 1) (n + 4) (n + 7)

= (n + 1) (n + 7) [\frac{n}{4} (n + 6) + n + 4]

= \frac{(n + 1) (n + 7)}{4} (n^{2} + 10 n + 16)

= \frac{(n + 1) (n + 7)}{4} (n + 2) (n + 8)

= \frac{(n + 1)}{4} (n + 2) (n + 7) (n + 8)

\dots Conclusion \dots