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Factorial identity used in an induction proof of a summation formula

Factorial identity used in an induction proof of a summation formula

8 marks
  1. Show that \(\displaystyle\frac{k+1}{(k+2)!}-\frac{k^2+3k+1}{(k+3)!}=\frac{k+2}{(k+3)!}\).[1]
  2. Use mathematical induction to prove that, for every positive integer \(n\),
    \(\displaystyle\sum_{r=1}^{n}\frac{r^2+r-1}{(r+2)!}=\frac12-\frac{n+1}{(n+2)!}\)[7]

Solution:

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Answer:(a) shown (b) proven by mathematical induction for all \(n\in\mathbb{Z}^+\)

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