Question

in a geometric sequence, a11=72 and a13=1/2 use the geometric mean to find the value of a12

Answer 1

\[
\frac{a_{13}}{a_{11}} = \frac{ar^{13-1}}{ar^{11-1}} =r^2
\]

Answer 2

So \[
r = \sqrt{a_{13}/a_{11}}
\]

Answer 3

\[
a_{13} = ar^{13-1} = arr^{12-1} = ra_{12}
\]

Answer 4

Therefore, \[
a_{12} = \frac{a_{13}}{r}
\]

Answer 5

Well, I mean, you really have two equations here in the end...

Answer 6

Since \[
a_n=ar^{n-1}
\]

Answer 7

We get: \[
a_{11} = ar^{11-1}\implies 72 = ar^{10}\\
a_{13} = ar^{13-1}\implies \frac12 = ar^{12}
\]

Answer 8

Two equations, two variables.