write an explicit formula for $a_{n}$, the $n^{th}$ term of the sequence 75, 15, 3, ....

Explanation:

Step1: Identify the type of sequence

This is a geometric sequence since there is a common - ratio between consecutive terms. To find the common ratio $r$, divide the second term by the first term. $r=\frac{a_{2}}{a_{1}}=\frac{15}{75}=\frac{1}{5}$.

Step2: Recall the formula for the $n$th term of a geometric sequence

The formula for the $n$th term of a geometric sequence is $a_{n}=a_{1}r^{n - 1}$, where $a_{1}$ is the first term and $r$ is the common ratio.

Step3: Substitute the values of $a_{1}$ and $r$ into the formula

Here, $a_{1}=75$ and $r = \frac{1}{5}$. So, $a_{n}=75\times(\frac{1}{5})^{n - 1}$.

Answer:

$a_{n}=75\times(\frac{1}{5})^{n - 1}$