Question

Find the first three terms of an infinitely decreasing geometric progression whose sum is $1.6$ and the second term is $-0.5$.

Answer 1

Given $a_2=-0.5$

$⟹a_1=\frac{-0.5}{r},a_3=-0.5r$

$\frac{-0.5}{r}-0.5-0.5r=1.6⟹\frac{r^2+1}{r}=-4.2⟹5r^2+21r+5=0⟹r=\frac{-21\pm \sqrt{321}}{10}=-0.308$

The three numbers are $0.154,-0.5,1.621$