Finding the nth Term of a Geometric Sequence

Property Let $A(n)$ equal the $n$th term of a geometric sequence, then $$A(n) = ar^{n 1}$$ where $a$ is the first term of the sequence and $r$ is the common ratio.

Explanation Want to jump to any term in a sequence without listing them all? This formula is your mathematical time machine! Just plug in the first term ($a$), the common ratio ($r$), and the term number you want ($n$). Remember the tricky part: the exponent is always one less than the term number ($n 1$) because the first term doesn't get multiplied yet!

Examples Find the 5th term if $a=6$ and $r= 2$: $A(5) = 6( 2)^{5 1} = 6( 2)^4 = 6(16) = 96$. Find the 8th term of $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ...$. Here $a=\frac{1}{2}$ and $r=\frac{1}{2}$. $A(8) = \frac{1}{2}(\frac{1}{2})^{8 1} = (\frac{1}{2})^8 = \frac{1}{256}$. A ball's first bounce is 1.5 yards high ($a=1.5$), and each bounce is 80% of the last ($r=0.8$). The 5th bounce height is $A(5) = 1.5(0.8)^{5 1} \approx 0.61$ yards.