First slide

Geometric progression

Question

If $(x) < 1,$ and $r$ th term of a series is $1 + x$ $+ x^{2} + \dots + x^{r - 1},$ then sum to $n$ terms of the series is

Moderate

A

$\frac{n + (n + 1) x - x^{n + 1}}{(1 - x)^{2}}$
B

$\frac{n - (n + 1) x + x^{n + 1}}{(1 - x)^{2}}$
C

$\frac{(n + 1) x - x^{n + 1} - n}{(1 - x)^{2}}$
D

$\frac{n - (n + 1) x - x^{n + 1}}{(1 - x)^{2}}$

Solution

$t_{r} = 1 + x + x^{2} + \dots x^{r - 1} = \frac{1 - x^{r}}{1 - x}$
Now, $\sum_{r = 1}^{n} t_{r} = \sum_{r = 1}^{n} (\frac{1 - x^{r}}{1 - x})$
$= \frac{n}{1 - x} - \frac{1}{1 - x} \sum_{r = 1}^{n} x^{r}$
$\begin{array}{l} = \frac{n}{1 - x} - \frac{1}{1 - x} \frac{x (1 - x^{n})}{1 - x} \\ = \frac{1}{(1 - x)^{2}} [n - n x - x + x^{n + 1}] \\ = \frac{1}{(1 - x)^{2}} [n - (n + 1) x + x^{n + 1}] \end{array}$

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