Evaluate: \text{begin}array l 60,

Expression: $\begin{array} { l }60,& 30,& 15,& \frac{ 15 }{ 2 },& …\end{array}$

Calculate the ratio between each pair of consecutive terms

$\begin{array} { l }r=\frac{ 30 }{ 60 },\\r=\frac{ 15 }{ 30 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Cancel out the common factor $30$

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 15 }{ 30 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Cancel out the common factor $15$

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Simplify the complex fraction

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 }\end{array}$

Since the ratio between each pair of consecutive terms is the same, the sequence is geometric and the common ratio is $r=\frac{ 1 }{ 2 }$

$r=\frac{ 1 }{ 2 }$

Substitute the first term $a_1=60$ and the common ratio $r=\frac{ 1 }{ 2 }$ into the formula for the $n$-th term of a geometric sequence, $a_n=a_1{r}^{n-1}$

$a_n=60 \times {\left( \frac{ 1 }{ 2 } \right)}^{n-1}$