Find $\sum_{r=0}^{k}\left(2r + 5 - 3^{1-r}\right)$ in terms of $k$, simplifying your answer.[4]

Question

Find $\sum_{r=0}^{k}\left(2r + 5 - 3^{1-r}\right)$ in terms of $k$, simplifying your answer.[4] Using the result in part (i), find $\sum_{r=0}^{k}\left(2r + 9 - 3^{-r-1}\right)$ in terms of $k$.[3]

Accepted Answer

(i) $(k+1)(k+5)-\dfrac92\left(1-\dfrac1{3^{k+1}}\right)$ (ii) $(k+3)(k+7)+\dfrac9{2\cdot3^{k+3}}-\dfrac{25}2$

Sequence and Series - Sigma Notation By Substitution

Sequence and Series - Sigma Notation By Substitution

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