# Arithmetic Sequence: How to Calculate Total Commissions Earned by a Salesperson

## How to determine the number of cars a salesperson must sell to earn $4,200 in commissions?

Given the following data:

First commission = $300.

Second commission = $400.

Common difference = $100.

Total commission = $4,200.

## Answer:

The equation that represents the number of cars a salesperson must sell to earn $4,200 in commissions is:

\[4200 = \frac{n}{2}(2(300) + (n-1)100)\]

To calculate the total commissions earned by a salesperson based on an arithmetic sequence, we can use the formula for the sum of an arithmetic sequence:

Where:

- \(d\) is the common difference.

- \(a_1\) is the first term of the arithmetic sequence.

- \(n\) is the total number of terms.

By substituting the given parameters into the formula, we get:

\[4200 = \frac{n}{2}(2(300) + (n-1)100)\]Based on the formula and calculation, the salesperson must sell 7 cars to earn $4,200 in commissions. This is achieved by increasing the commission by $100 for each additional car sold in a week. Therefore, to reach the total commission amount, the salesperson needs to sell cars that correspond to an arithmetic sequence.