Question

write an equation to describe the sequence below. use n to represent the position of a term in the sequence, where n = 1 for the first term.
31, 62/5, 124/25, ...
write your answer using proper fractions, improper fractions, and integers.
a_n = ( )^{n - 1}

Explanation:

Step1: Identify the first - term

The first - term $a_1$ of the sequence is $31$.
$a_1 = 31$

Step2: Find the common ratio

To find the common ratio $r$, divide the second term by the first term. The second term is $\frac{62}{5}$, and the first term is $31$. So, $r=\frac{\frac{62}{5}}{31}=\frac{62}{5}\times\frac{1}{31}=\frac{2}{5}$.

Step3: Write the general formula for a geometric sequence

The general formula for a geometric sequence is $a_n=a_1r^{n - 1}$. Substituting $a_1 = 31$ and $r=\frac{2}{5}$ into the formula, we get $a_n=31\times(\frac{2}{5})^{n - 1}$.

Answer:

$a_n = 31\times(\frac{2}{5})^{n - 1}$