The sum of the series 1+43+109+2827+ up to n terms is:

Question

The sum of the series $1+\dfrac43+\dfrac{10}{9}+\dfrac{28}{27}+\cdots$ up to $n$ terms is:

Accepted Answer

Writing each term after the first as $1+\left(\dfrac13\right)^{k}$ (e.g. $\tfrac43=1+\tfrac13,\ \tfrac{10}9=1+\tfrac19,\ \tfrac{28}{27}=1+\tfrac{1}{27}$):

$$S_n=1+\sum_{k=1}^{n-1}\left[1+\left(\frac13\right)^k\right]=n+\sum_{k=1}^{n-1}\left(\frac13\right)^k=n+\frac12\left[1-\left(\frac13\right)^{n-1}\right]=n+\frac12-\frac{1}{2\cdot3^{n-1}}.$$

(Check $n=1$: gives $1$ ; $n=2$: gives $7/3=1+4/3$ .)

Solutions · 1