Question

write an equation to describe the sequence below, and then find the 4th term. use n to represent the position of a term in the sequence, where n = 1 for the first term. -30, -45/2, -135/8, ... write your answer using proper fractions, improper fractions, and integers. a_n = ( )^{n - 1} a_4 =

Explanation:

Step1: Identify the first - term and common ratio

The first - term $a_1=-30$. To find the common ratio $r$, divide the second term by the first term. $r=\frac{\frac{-45}{2}}{-30}=\frac{-45}{2}\times\frac{-1}{30}=\frac{3}{4}$.

Step2: Write the general formula for the geometric sequence

The general formula for a geometric sequence is $a_n=a_1r^{n - 1}$. Substituting $a_1=-30$ and $r = \frac{3}{4}$, we get $a_n=-30\times(\frac{3}{4})^{n - 1}$.

Step3: Find the 4th term

Substitute $n = 4$ into the formula $a_n=-30\times(\frac{3}{4})^{n - 1}$. Then $a_4=-30\times(\frac{3}{4})^{4 - 1}=-30\times(\frac{3}{4})^3=-30\times\frac{27}{64}=-\frac{405}{32}$.

Answer:

$a_n=-30\times(\frac{3}{4})^{n - 1}$
$a_4=-\frac{405}{32}$