# The sum of the first four terms of a geometric series is 130 and its common ratio is 1.5. What is the first term in the series?

Mar 24, 2016

First four terms of the series are $16 , 24 , 36 , 54$

#### Explanation:

As common ratio is $1.5$ or $\frac{3}{2}$, let the first four terms of the geometric series be $a , \frac{3}{2} \cdot a , {\left(\frac{3}{2}\right)}^{2} a , {\left(\frac{3}{2}\right)}^{3} a$ or

$a , \frac{3}{2} a , \frac{9}{4} a$ and $\frac{27}{8} a$.

Now there sum is $130$, hence

$a + \frac{3 a}{2} + \frac{9 a}{4} + \frac{27 a}{8} = 130$ or

$\frac{8 a + 12 a + 18 a + 27 a}{8} = 130$ or

$\frac{65 a}{8} = 130$ or $a = 2 \cancel{130} \times \frac{8}{1 \cancel{65}} = 16$

Hence first four terms of the series are

$16 , 16 \times \frac{3}{2} , 16 \times \frac{9}{4} , 16 \times \frac{27}{8}$ or

$16 , 24 , 36 , 54$