Question

1. higher order thinking in case of an emergency, the school has a calling list so that everyone is called in the least amount of time. each of the first 3 people on the list calls another 3 people on the list. then, each of the people in the second group calls another 3 people on the list, and so on. the fifth group of people will make 243 calls. is this statement accurate? explain.

Explanation:

Step1: Identify the pattern

This is a geometric - growth problem. The first group has 3 people, and each subsequent group has 3 times as many call - makers as the previous group. The number of call - makers in the $n$th group forms a geometric sequence with first term $a_1 = 3$ and common ratio $r = 3$. The formula for the $n$th term of a geometric sequence is $a_n=a_1\times r^{n - 1}$.

Step2: Substitute values

We want to find the number of call - makers in the 5th group. Here, $a_1 = 3$, $r = 3$, and $n = 5$. Substitute these values into the formula $a_n=a_1\times r^{n - 1}$. So $a_5=3\times3^{5 - 1}$.

Step3: Simplify the expression

Using the rule of exponents $a^m\times a^n=a^{m + n}$, we have $a_5=3\times3^{4}=3^{1 + 4}=3^5$. And $3^5=3\times3\times3\times3\times3 = 243$.

Answer:

Yes, the statement is accurate. The number of calls made by the fifth group is 243 since using the formula for the $n$th term of a geometric sequence $a_n=a_1\times r^{n - 1}$ with $a_1 = 3$, $r = 3$, and $n = 5$ gives $a_5=3^5=243$.