Finding the $n^{\text{th}}$ Term of a Geometric Sequence

Example 1:

Find the $6^{\text{th}}$ term in the geometric sequence $3,12,48,\mathrm{...}$ .

$\begin{array}{l}{a}_1=3,r=\frac{12}{3}=4\\ a_6=3\cdot 4^{6-1}=3\cdot 4^5=3072\end{array}$

Example 2:

Find the $7^{\text{th}}$ term for the geometric sequence in which $a_2=24$ and $a_5=3$ .

Substitute $24$ for $a_2$ and $3$ for $a_5$ in the formula

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

$\begin{array}{l}{a}_2=a_1\cdot r^{2-1}\to 24=a_{1}r\\ a_5=a_1\cdot r^{5-1}\to 3=a_1{r}^4\end{array}$

Solve the firstequation for $a_1$ : $a_1=\frac{24}{r}$

Substitute this expression for $a_1$ in the second equation and solve for $r$ .

$\begin{array}{l}3=\frac{24}{r}\cdot r^4\\ 3=24r^3\\ \frac{1}{8}=r^3\text{ so }r=\frac{1}{2}\end{array}$

Substitute for $r$ in the first equation and solve for $a_1$ .

$\begin{array}{l}24=a_1\left(\frac{1}{2}\right)\\ 48=a_1\end{array}$

Now use the formula to find $a_7$ .

$a_7=48{\left(\frac{1}{2}\right)}^{7-1}=48\cdot \frac{1}{64}=\frac{3}{4}$