Question

The sum of the first two terms of a geometric progression is 9 and the sum to infinity is 25. If the common ratio is positive, find the common ratio and the first term.   

Answer 1

$$a+ar=9$$

$$S=\dfrac{a}{1-r}=25, r>0$$

$$\dfrac{9}{1+r}=25(1-r)$$

$$25r^2=16$$

$$r_1=-\dfrac{4}{5}, r_2=\dfrac{4}{5}$$

Since $r>0,$ we take $r=\dfrac{4}{5}.$

$$a=\dfrac{9}{1+\dfrac{4}{5}}=5$$

$a=5, r=\dfrac{4}{5}$