The series is a, ar, ar2, ar3, ..., arn-1.
Sn=a(rn-1)/(r-1).
S4=10S2.
S4=a(r4-1)/(r-1) and S2=a(r2-1)/(r-1).
Therefore:
a(r4-1)/(r-1)=10a(r2-1)/(r-1).
r4-1=10r2-10,
r4-10r2+9=0=(r2-1)(r2-9)=(r-1)(r+1)(r-3)(r+3).
Since r>1, r=3 is the only solution.
Let's check:
Let a=1, then the series becomes 1, 3, 9, 27, 81, ....
S4=1+3+9+27=40 and S2=1+3=4 so S4=10S2, confirming r=3.