# Math Help - Sequence

1. ## Sequence

Find the third element of a geometric sequence who fifth element is 81 and whose ninth element is 16. Help please!

2. Originally Posted by reiward
Find the third element of a geometric sequence who fifth element is 81 and whose ninth element is 16. Help please!
Hi reiward,

You should have been given the formula for finding the nth term of a geometric sequence.

$a_n=a_1 \cdot r^{n-1}$

$r$ = common ratio

Your first element ( $a_1$) will be 81.

$n = 5$ because there are five elements in all between the 5th element and the 9th element, inclusive.

$a_n = 16$

Solve for $r$

Now that you have found the common ratio, substitute back into the original formula and solve for $a_3$.

This time there are 3 elements from the 3rd to the 5th elements, inclusive. So, $n=3$

Remember $a_5=81$

$a_5=a_3 \cdot r^{3-1}$

3. Hello, reiward!

Use the formula for the $n^{th}$ term: . $a_n \:=\:ar^{n-1}$

Find the third element of a geometric sequence who fifth element is 81
and whose ninth element is 16.
We are given: . $\begin{array}{ccccccc}a_5 &=& ar^4 &=& 81 & [1] \\ a_9 &=& ar^8 &=& 16 & [2] \end{array}$

Divide [2] by [1]: . $\frac{ar^8}{ar^4} \:=\:\frac{16}{81} \quad\Rightarrow\quad r^4 \:=\:\frac{16}{81} \quad\Rightarrow\quad r \:=\:\pm\frac{2}{3}$

Substitute into [1]: . $a\left(\pm\frac{2}{3}\right)^4 \:=\:81 \quad\Rightarrow\quad \frac{16}{81}a\:=\:81 \quad\Rightarrow\quad a \:=\:\frac{6561}{16}$

Therefore: . $a_3\;=\;ar^2 \;=\;\frac{6561}{16}\left(\pm\frac{2}{3}\right)^2 \;=\;\frac{729}{4}$