Question

geometric series consists of even number of terms . sum of all terms is 3 times the sum of the odd terms. find the common ratio​

Answer 1

Answer:

Geometric series has even number of terms .

Let the G.P have n terms .

We know that the Sum of G.P is given by $a\dfrac{r^n-1}{r-1}$ .

Now the odd terms , there will be $\dfrac{n}{2}$ terms .

Common ratio will be $r^2$

Sum of such terms will be $a(\dfrac{r^{2\times n/2}-1}{r^2-1})$

Given :

$a(\dfrac{r^n-1}{r-1})=3a(\dfrac{r^{2\times n/2}-1}{r^2-1})\\\\\implies \dfrac{r^n-1}{r-1}=3(\dfrac{r^n-1}{(r-1)(r+1)})\\\\\implies 1=\dfrac{3}{r+1}\\\\\implies r+1=3\\\\\implies r=3-1\\\\\implies r=2$

Common ratio is 2 .

Step-by-step explanation

First G.P is $a,ar,ar^2......ar^n$

Second G.P is $a,ar^3,ar^5...ar^{n-1}$

The sum of second G.P is thrice the first .

Use the sum of G.P as :

$S=a\dfrac{r^n-1}{r-1}$