Find the sum to indicated number of terms in each of the geometric progressions in $x^{3}, x^{5}, x^{7}, . . . . . n t e r m s (i f x \neq \pm 1) .$

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Solution

Here, $a = x^{3}$ and $r = \frac{x^{5}}{x^{3}} = x^{2}$

We know that $S_{n} = \frac{a (1 - r^{n})}{1 - r}$

$∴ S_{n} = \frac{x^{3} [1 - (x^{2})^{n}]}{1 - x^{2}} = \frac{x^{3}}{1 - x^{2}} [1 - x^{2 n}]$

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Find the sum to indicated number of terms in each of the geometric progressions in x3,x5,x7,.....n terms(if x ≠±1).

Here, a=x3 and r=x5x3=x2 We know that Sn=a(1−rn)1−r ∴Sn=x3[1−(x2)n]1−x2=x31−x2[1−x2n]

Find the sum to indicated number of terms in each of the geometric progressions in $x^{3}, x^{5}, x^{7}, . . . . . n t e r m s (i f x \neq \pm 1) .$

Here, $a = x^{3}$ and $r = \frac{x^{5}}{x^{3}} = x^{2}$

We know that $S_{n} = \frac{a (1 - r^{n})}{1 - r}$

$∴ S_{n} = \frac{x^{3} [1 - (x^{2})^{n}]}{1 - x^{2}} = \frac{x^{3}}{1 - x^{2}} [1 - x^{2 n}]$