The first year of a charity walk event had an attendance of $500$ . The attendance $y$ increases by $5 \%$ each year. How many people will attend in the $10$ th year? Round your answer to the nearest person. $\newline$

Full solution

Q. The first year of a charity walk event had an attendance of

500

. The attendance

y

increases by

5 \%

each year. How many people will attend in the

10

th year? Round your answer to the nearest person.

\newline

Determine Formula: Determine the formula to calculate the attendance in the $10^{\text{th}}$ year. $\newline$ The attendance increases by a fixed percentage each year, which is a geometric sequence. The formula for the future value of a quantity growing at a fixed percentage rate is given by: $\newline$ Future Value $=$ Present Value $\times (1 + \text{growth rate})^{\text{number of periods}}$ $\newline$ Here, the Present Value is the initial attendance, the growth rate is the annual increase in attendance, and the number of periods is the number of years.
Plug Values: Plug in the values into the formula. $\newline$ Present Value (initial attendance): $500$ $\newline$ Growth rate (annual increase): $5\%$ or $0.05$ $\newline$ Number of periods (years): $10$ $\newline$ Future Value = $500 \times (1 + 0.05)^{10}$
Calculate Future Value: Calculate the future value using the formula. $\newline$ Future Value = $500 \times (1 + 0.05)^{10}$ $\newline$ Future Value = $500 \times (1.05)^{10}$ $\newline$ Future Value = $500 \times 1.628894626777442$ $\newline$ Future Value $\approx 814.447313388721$
Round Answer: Round the answer to the nearest person. $\newline$ Since we cannot have a fraction of a person attending, we round the number to the nearest whole number. $\newline$ Future Value $\approx 814$ people