Question

find a formula for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively

Answer 1

\[a_n=a_1r^{(n-1)}\rightarrow \frac{ a_5 }{ a_2 }=r^3=-\frac{ 567 }{ 21 }\rightarrow r=\sqrt[3]{-\frac{ 567}{ 21 }}=-3\]\[a_1=\frac{ a_2 }{ r }=\frac{ -21 }{ -3 }=7 \rightarrow a_n=7·(-3)^{(n-1)}\]

Answer 2

Good job. The only thing i would add in is \(\large \frac{a_5}{a_2}= a_3 = r^3\)

Answer 3

But why do you say that \[\frac{ a_5 }{ a_2}=a_3\] ???? Where is this relation coming from?

Answer 4

\[a_3=7·(-3)^2=7·9=63\] and \[a_2=-21\rightarrow a_2·a_3=-21·63 \neq 267\]