Question

The sum of an infinite geometric progression is $243$ and the sum of its first five terms is $275$. Find the progression.

Answer 1

$a+ar+a{r}^2+.....\infty =243⟹\frac{a}{1-r}=243$

$a+ar+a{r}^2+a{r}^3+a{r}^4=275⟹\frac{a(1-r^5)}{1-r}=275⟹1-r^5=\frac{275}{243}⟹r^5=-\frac{32}{243}⟹r=-\frac{2}{3}$

$a=243(1-r)=243\left(\frac{4}{3}\right)=324$

the progression is $324,-216,144,-96,....$