A new shopping mall records $150$ total shoppers on their first day of business. Each day after that, the number of shoppers is $15\%$ more than the number of shoppers the day before. $\newline$ Which expression gives the total number of shoppers in the first $n$ days of business? $\newline$ Choose $1$ answer: $\newline$ (A) $1.15\left(\frac{1-150^n}{1-150}\right)$ $\newline$ (B) $0.85\left(\frac{1-150^n}{1-150}\right)$ $\newline$ (C) $150\left(\frac{1-1.15^n}{1-1.15}\right)$ $\newline$ (D) $150\left(\frac{1-0.85^n}{1-0.85}\right)$

Full solution

Q. A new shopping mall records

150

total shoppers on their first day of business. Each day after that, the number of shoppers is

15\%

more than the number of shoppers the day before.

\newline

Which expression gives the total number of shoppers in the first

n

days of business?

\newline

Choose

1

answer:

\newline

(A)

1.15\left(\frac{1-150^n}{1-150}\right)

\newline

(B)

0.85\left(\frac{1-150^n}{1-150}\right)

\newline

(C)

150\left(\frac{1-1.15^n}{1-1.15}\right)

\newline

(D)

150\left(\frac{1-0.85^n}{1-0.85}\right)

Calculate Daily Increase: First day shoppers are $150$ . Each subsequent day increases by $15\%$ , so multiply by $1.15$ for each day after the first.
Geometric Series Formula: The total number of shoppers over $n$ days is a geometric series with the first term $a = 150$ and common ratio $r = 1.15$ .
Calculate Total Shoppers: The sum of the first $n$ terms of a geometric series is given by $S_n = \frac{a(1-r^n)}{1-r}$ .
Substitute Values: Substitute $a = 150$ and $r = 1.15$ into the formula to get the expression for the total number of shoppers: $S_n = 150\left(\frac{1-1.15^n}{1-1.15}\right)$ .
Match with Answer Choices: Check the answer choices to find the expression that matches our calculation.