The sum of the first 20 terms of the series 1+32+74+158+3116+ is:

Question

The sum of the first 20 terms of the series $1+\dfrac32+\dfrac74+\dfrac{15}{8}+\dfrac{31}{16}+\cdots$ is:

Accepted Answer

General term (from $r=0$): $T_r = 2-\dfrac{1}{2^r}$. Excluding the first term and summing the remaining 19:

$$\sum_{r=1}^{19}\left(2-\frac{1}{2^r}\right) = 38 - \left(1-\frac{1}{2^{19}}\right) = 37+\frac{1}{2^{19}}.$$

Adding the first term ($=1$): $S_{20} = 38+\dfrac{1}{2^{19}}.$

Solutions · 1