📌 Example 1: **Tax Rebates and the Multiplier Effect**

A tax rebate that returns a certain amount of money to taxpayers can have a total effect on the economy that is many times this amount. In economics, this phenomenon is called the **multiplier effect**. Suppose, for example, that the government reduces taxes so that each consumer has $2000 more income. The government assumes that each person will spend 70% of this (= $1400). The individuals and businesses receiving this $1400 in turn spend 70% of it (= $980), creating extra income for other people to spend, and so on. Determine the total amount spent on consumer goods from the initial $2000 tax rebate.

✍ Solution:

The total amount spent is given by the infinite geometric series

^{2}+⋯

=1400+980+686 +⋯.

The first term is 1400:

*a*

_{1}=1400. The common ratio is 70%, or 0.7:

*r*=0.7. Because

*r*=0.7, the condition that |

*r*|<1 is met. Thus, we can use our formula to find the sum. Therefore,

This means that the total amount spent on consumer goods from the initial $2000 rebate is approximately $4667.

📌 Question 1: **Computing a Lifetime Salary**

A union contract specifies that each worker will receive a 5% pay increase each year for the next 30 years. One worker is paid $20,000 the first year. What is this person’s total lifetime salary over a 30-year period?

✍ Solution:

The salary for the first year is $20,000. With a 5% raise, the second—year salary is computed as follows:

Salary for year 2=20,000+20,000(0.05)=20,000(1+0.05)=20,000(1.05).

Each year, the salary is 1.05 times what it was in the previous year. Thus, the salary for year 3 is 1.05 times 20,000(1.05), or 20,000(1.05)^{2}. The salaries for the first five years are given in the table.

Yearly Salaries

The numbers in the bottom row form a geometric sequence with

*a*

_{1}=20,000 and

*r*=1.05. To find the total salary over 30 years, we use the formula for the sum of the first n terms of a geometric sequence, with n=30.

Use a calculator.

The total salary over the 30-year period is approximately $1,328,777..

📌 Example 2 (Allowance). Danielle’s parents have offered her two different options to earn her allowance for a 9-week period over the summer. She can either get paid $30 each week or $1 the first week, $2 for the second week, $4 for the third week, and so on.

a. Does the second option form a geometric sequence? Explain.

b. Which option should Danielle choose? Explain.

✍ Solution:

a. Calculate the common ratio.

There is a common ratio of 2. So, the second option does form a geometric sequence.

b. Calculate how much Danielle would earn with each option.

Option 1

Option 2

In nine weeks, Danielle would earn $270 with the first option and $511 with the second option. So, she should choose the second option.