Question

Taye recruits two people to be election campaign volunteers. The next week he ask each of those volunteers to recruit two more campaign volunteers. He wants all new volunteers each week to recruit two more volunteers.
a. Determine a method to calculate the number of volunteers in any given week. Use that method to
calculate the number of volunteers recruited for each of the first 5 weeks.
b. Taye wants to recruit 150 volunteers by election day. During which week can some of the
volunteers stop recruiting new volunteers? Explain your reasoning.

Answer 1

Using a geometric sequence, it is found that:

a) For the first 5 weeks, the numbers are: 2, 4, 8, 16, 32

b) Hence some of the volunteers can stop recruiting new volunteers during the 7th week.

What is a geometric sequence?

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

The nth term of a geometric sequence is given by:

$a_n = a_1q^{n-1}$

In which $a_1$ is the first term.

For this problem, on the first week there are 2 people, and since each new people involve two more, the common ratio is of 2, hence the parameters for the geometric sequence are:

$a_1 = 2, q = 2$

Hence the number of people after n weeks is:

$a_n = 2 \times 2^{n - 1}$

For the first 5 weeks, the numbers are:

• $a_1 = 2 \times 2^{1 - 1} = 2$

• $a_2 = 2 \times 2^{2 - 1} = 4$

• $a_3 = 2 \times 2^{3 - 1} = 8$

• $a_4 = 2 \times 2^{4 - 1} = 16$

• $a_5 = 2 \times 2^{5 - 1} = 32$

There will be 150 volunteers when $a_n = 150$, hence:

$2 \times 2^{n - 1} = 150$

$2^{n-1} = 75$

$\frac{2^n}{2} = 75$

$2^n = 150$

$n = 7...$

The above is because 2^7 = 128 and 2^8 = 256, and 128 < 150 < 256.

Hence some of the volunteers can stop recruiting new volunteers during the 7th week.

More can be learned about geometric sequences at brainly.com/question/11847927

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