Show that $\displaystyle \frac{1}{(k+2)!}-\frac{k+1}{(k+3)!}=\frac{2}{(k+3)!}$. Prove by induction

Question

Show that $\displaystyle \frac{1}{(k+2)!}-\frac{k+1}{(k+3)!}=\frac{2}{(k+3)!}$. Prove by induction that for all positive integers $n$, $\displaystyle \sum_{r=1}^{n}\frac{r\cdot2^r}{(r+2)!}=1-\frac{2^{n+1}}{(n+2)!}$

Accepted Answer

The identity holds, and the stated summation formula is true for every positive integer $n$ by induction.

Solution:

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