First slide

Geometric progression

Question

$The sum of the first 20 terms of the series 1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots is$

Difficult

A

$38 + \frac{1}{2^{20}}$
B

$39 + \frac{1}{2^{19}}$
C

$39 + \frac{1}{2^{20}}$
D

$38 + \frac{1}{2^{19}}$

Solution

$General term of the given series \frac{{2.2}^{r} - 1}{2^{r}}, r \geq 0$
$\begin{aligned} Required sum = 1 + \sum_{i = 1}^{19} \frac{{2.2}^{r} - 1}{2^{r}} \\ Now \sum_{r = 1}^{19} \frac{{2.2}^{r} - 1}{2^{r}} = \sum_{s = 1}^{19} (2 - \frac{1}{2^{r}}) \end{aligned}$
$\begin{array}{l} = 2 \times 19 - \frac{\frac{1}{2} (1 - \frac{1}{2^{19}})}{1 - 1 / 2} = 38 + \frac{1}{2^{19}} - 1 (\sin c e s u m o f n t e r m s i n G . P = \frac{a (1 - r^{n})}{1 - r}) \\ = 37 + \frac{1}{2^{19}} \\ Required sum = 1 + 37 + \frac{1}{2^{19}} = 38 + \frac{1}{2^{19}} \end{array}$

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