Name: Sum of First n Odd Numbers
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: The odd numbers sequence also forms an A.P. whose common difference is 2. In this video, we will learn to find the formula for the sum of first n odd numbers...

Question

The second term in a geometric series is

180

and the sixth term is

\frac{20}{9} .

Find the sum to Infinity of the series.

Accepted Answer

It is given that $a_{2} = 180, a_{6} = \frac{20}{9}$

To find sum to Infinity of the geometric series.

We know that $a_{n} = a r^{n - 1}$

$∴ a_{2} = a r$

$\Rightarrow 180 = a r - (1)$

and $a_{6} = a r^{5}$

$\Rightarrow \frac{20}{9} = a r^{5} - (2)$

Now, divide $(2)$ by $(1)$ , we get:

$\frac{a r^{3}}{a r} = \frac{\frac{20}{9}}{180}$

$\Rightarrow r^{4} = \frac{1}{81}$

$\Rightarrow r = \pm \frac{1}{3}$

Putting the value of $r$ in $(1)$ , we get:

$\pm \frac{1}{3} a = 180$

$\Rightarrow a = \pm 540$

Now, when $r = \frac{1}{3}, a = 540$ then

$S_{\infty} = \frac{a}{1 - r}$

$\Rightarrow S_{\infty} = \frac{540}{1 - \frac{1}{3}}$

$\Rightarrow S_{\infty} = 540 \times \frac{3}{2}$

$\Rightarrow S_{\infty} = 810$

and when $r = - \frac{1}{3}, a = - 540$ then

$S_{\infty} = \frac{a}{1 - r}$

$\Rightarrow S_{\infty} = \frac{- 540}{1 + \frac{1}{3}}$

$\Rightarrow S_{\infty} = - 540 \times \frac{3}{4}$

$\Rightarrow S_{\infty} = - 405$

The second term in a geometric series is $180$ and the sixth term is $\frac{20}{9} .$ Find the sum to Infinity of the series.

Important Questions on From Patterns to Generalizations: Sequences and Series

Learn and Prepare for any exam you want !

The second term in a geometric series is 180 and the sixth term is 209. Find the sum to Infinity of the series.

Important Questions on From Patterns to Generalizations: Sequences and Series

Learn and Prepare for any exam you want !

The second term in a geometric series is $180$ and the sixth term is $\frac{20}{9} .$ Find the sum to Infinity of the series.