Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-A small business owner made $50,000 the first year he owned his store and made an additional 2% over the previous year in each subsequent year. Find how much he made during his fourth year of business. Find his total earnings during the first four years. (Round to the nearest cent, if necessary.)
A) $53,060.40; $206,080.40
B) $0.40; $51,020.40
C) $86,400.00; $268,400.00
D) $103,680.00; $372,080.00

Question

Quizplus · Accepted Answer

The earnings each year form a geometric sequence with the first term $50,000 and a common ratio of 1.02 (reflecting a 2% increase each year). The fourth year's earnings can be found using the formula for the nth term of a geometric sequence, $a_n = a_1 \cdot r^{(n-1)}$, where $a_1 = 50,000$, $r = 1.02$, and $n = 4$. This calculation gives the earnings for the fourth year as approximately $53,060.40. The total earnings over the four years can be found using the formula for the sum of the first n terms of a geometric sequence, $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, which gives approximately $206,080.40 for the first four years.

Find the Indicated Sum