Buffy's Beverages has increasing profits each week during the hot summer months. Buffy finds that each week during the summer his profits increase by 22% over what they were the previous week. In his first week of business, his profits were $282.

How much money will he make in all over 7 weeks?

Enter your answer, rounded to the nearest cent.

To calculate how much money Buffy will make in total over 7 weeks, we can use the formula for the sum of the first n terms of a geometric series

[tex]\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}[/tex]

In his first week of business, Buffy's profits were $282.

Therefore, a = 282.

As his profits increase by 22% each week, his revenue is 122% of the previous week. As a decimal, 122% = 1.22.

Therefore, r = 1.22.

We want to find the total amount he will make over 7 weeks.

Therefore, n = 7.

Substitute the values of a, r and n into the formula and solve:

[tex]\begin{aligned}S_7&=\dfrac{282(1-1.22^7)}{1-1.22}\\\\&=\dfrac{282(1-4.0227108...)}{-0.22}\\\\&=\dfrac{282(-3.0227108...)}{-0.22}\\\\&=\dfrac{-0.852.404454...}{-0.22}\\\\&=3874.5657\end{aligned}[/tex]

Therefore, the total amount Buffy will make over 7 weeks is $3,874.57 to the nearest cent.