Let $S_n$ denote the sum of the first $n$ terms of the geometric sequence $\{a_n\}$. If $S_4=-5$, $S_6=21S_2$, then $S_8=$ A. 120 B. 85 C. $-85$ D. $-120$
C By the properties of geometric sequences, $S_2$, $S_4-S_2$, $S_6-S_4$ form a geometric sequence. Therefore $(S_4-S_2)^2=S_2(S_6-S_4)$. Substituting $S_4=-5$ and $S_6=21S_2$ into the equation and solving gives $S_2=-1$ (rejected) or $\frac{5}{4}$. At this time, $S_6=\frac{105}{4}$. By the properties of geometric sequences, $S_4-S_2$, $S_6-S_4$, $S_8-S_6$ form a geometric sequence. Solving gives $S_8=-85$. Choose C.
Let $S_n$ denote the sum of the first $n$ terms of the geometric sequence $\{a_n\}$. If $S_4=-5$, $S_6=21S_2$, then $S_8=$\\
A. 120\\
B. 85\\
C. $-85$\\
D. $-120$