If the rth term of a series is 1 + x + x2 + .......+ xr – 1, then the sum of the first n terms is

Engineering

Mathematics

Geometric Progression and Its Properties

Summation of Series

polynomial function

Question

If the r^th term of a series is 1 + x + x² + .......+ x^{r – 1}, then the sum of the first n terms is

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

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

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

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

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$T_{r} = 1 + x + x^{2} + \dots \dots . . + x^{r - 1} = \frac{1 - x^{r}}{1 - x}$ , the sum of the G.P.

$\sum_{r = 1}^{n} t_{r} = \sum_{r = 1}^{n} \frac{1 - x^{r}}{1 - x} = \sum_{r = 1}^{n} \frac{1}{1 - x} - \frac{\sum_{r = 1}^{n} x^{r}}{1 - x} = \frac{n}{1 - x} - \frac{1}{1 - x} (x + x^{2} + \dots \dots + x^{n})$

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