Question

In a geometric sequence, if a3 = ‒5 and a6 = 40, determine a1, r, and an. Then write the first three terms of the sequence.

Answer 1

$a_4=ra_3$
$a_5=ra_4=r^2a_3$
$a_6=ra_5=r^3a_3$

$40=-5r^3\implies r^3=-8\implies r=-2$

$a_3=ra_2=r^2a_1$

$-5=(-2)^2a_1\implies a_1=-\dfrac54$

You know the first and third terms, $a_1$ and $a_3$, so you just need to find the second term $a_2$:

$a_2=ra_1\implies a_2=-2\left(-\dfrac54\right)=\dfrac52$