Question

If the G.P.'s 5, 10, 20, ...... and 1280, 640, 320, .... have their nth terms equal, find the value of n.

Answer 1

Given :

First term, a = 5

Common ratio, r = 2

$a_n=\left(5\right)(2{)}^{n-1}$ ............(i)

Similarly, $a_n=\left(1280\right){\left(\frac{1}{2}\right)}^{n-1}$ ........(ii)

From (1) and (2)

$\left(5\right)(2{)}^{n-1}=(1280){\left(\frac{1}{2}\right)}^{n-1}$

$\Rightarrow \frac{1}{256}={\left(\frac{1}{4}\right)}^{n-1}$

$\Rightarrow {\left(\frac{1}{4}\right)}^4={\left(\frac{1}{4}\right)}^{n-1}$

$\Rightarrow n-1=4$

$\Rightarrow n=5$