Question

question
currently, the number of people entering a shopping mall is 114. it is estimated that the number of people entering the shopping mall will decrease by 5% each hour. find the total number of people entering the shopping mall at the end of 5 hours.
select the correct answer below:
490
513
516
604

Explanation:

Step1: Identify the formula for a geometric - series

The number of people entering the mall each hour forms a geometric series with first - term \(a = 114\) and common ratio \(r=1 - 0.05=0.95\). The sum of the first \(n\) terms of a geometric series is \(S_n=\frac{a(1 - r^n)}{1 - r}\).

Step2: Substitute the values into the formula

Here, \(a = 114\), \(r = 0.95\), and \(n = 5\). So \(S_5=\frac{114\times(1 - 0.95^5)}{1 - 0.95}\).
First, calculate \(0.95^5=0.95\times0.95\times0.95\times0.95\times0.95 = 0.7737809375\).
Then \(1-0.95^5=1 - 0.7737809375 = 0.2262190625\).
\(1 - r=1 - 0.95 = 0.05\).
\(S_5=\frac{114\times0.2262190625}{0.05}\).
\(114\times0.2262190625 = 25.788973125\).
\(S_5=\frac{25.788973125}{0.05}=515.7794625\approx516\).

Answer:

516