Let a + ar + ar^2 + ar^3 + \dotsb be an infinite geometric series. The sum of the series is $4.$ The sum of the cubes of all the terms is $10.$ Find the common ratio.
a+ar+ar2+...=a1−r as long as r<1
a3+a3(r3)+a3(r3)2+...=a31−r3 as long as r^3<1
So:
a1−r=4
and
a31−r3=10
Can you take it from here?