Question

calculating a term in a geometric sequence
kaelyns renters insurance premium is $20 per month. after one year, the premium increases to $21 per month. assuming the premium increases by the same percentage each year, what will kaelyns insurance premium be in the fourth year?
use the geometric sequence formula and round to the nearest cent. enter your answer in the box.
$a_n = a_1 cdot r^{n - 1}$
rounded to the nearest cent, kaelyns premium will be $\square$ per month in the fourth year.

Explanation:

Step1: Find the common ratio \( r \)

The first term \( a_1 = 20 \) (premium in the first year), and the second term \( a_2 = 21 \) (premium in the second year). The common ratio \( r \) in a geometric sequence is \( r=\frac{a_{n + 1}}{a_n}\), so \( r=\frac{21}{20}=1.05 \).

Step2: Identify the term number \( n \)

We want the premium in the fourth year. The first year is \( n = 1 \), so the fourth year is \( n = 4 \). The geometric sequence formula is \( a_n=a_1\times r^{n - 1} \).

Step3: Calculate \( a_4 \)

Substitute \( a_1 = 20 \), \( r = 1.05 \), and \( n = 4 \) into the formula:
\( a_4=20\times(1.05)^{4 - 1}=20\times(1.05)^3 \)
First, calculate \( (1.05)^3=1.05\times1.05\times1.05 = 1.157625 \)
Then, \( 20\times1.157625 = 23.1525 \)
Rounding to the nearest cent (two decimal places), we get \( 23.15 \).

Answer:

\( 23.15 \)