The closed form for this sum, $$\sum_{r=1}^{n} (2r) 3^{n-r}=\frac{1}{2} \left(-3 + 3^{1 + n} - 2 n\right)$$ is quoted in my textbook. How would I derive it?
Asked
Active
Viewed 27 times
1 Answers
0
Let $S:=\sum_{r=1}^{n}r*3^{n-r}$. Multiply both sides by 3 and subtract $S$ from $3S$. Then, collect the like terms to get: $$2S=3^n+3^{n-1}+...+3-n$$ and $$S=\frac{1}{2}(\frac{3(3^n-1)}{2}-n)$$ Now multiply this by $2$ and get what you want, because initially I omitted the coefficient $2$ in the sum.
VIVID
- 11,604
-
Please check whether this answer provides additional value beyond the many answers to the original question linked to in a comment under the question. If so, please post it there instead to prevent fragmentation of answers across duplicates. – joriki Mar 28 '20 at 09:20