Using the result $\sum\limits_{r=1}^{n}{\frac{r}{{{2}^{r}}}}=2-\frac{n+2}{{{2}^{n}}}$, show that $\s

Question

Using the result $\sum\limits_{r=1}^{n}{\frac{r}{{{2}^{r}}}}=2-\frac{n+2}{{{2}^{n}}}$, show that $\sum\limits_{r=1}^{n}{\left( r-n \right)\left( {{2}^{-r}}+1 \right)}$ can be expressed in the form $C\left( 1-\frac{1}{{{2}^{n}}} \right)+Dn\left( n+1 \right)$, where $C$ and $D$ are constants to be determined. [4]

Accepted Answer

$2\left(1-\frac1{2^n}\right)-\frac12n(n+1)$, so $C=2,D=-\frac12$

2019 DHS Promo Q2

2019 DHS Promo Q2

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