Problem 59 - College Algebra

Question

A company offers a starting yearly salary of \(\$ 33,000\) with raises of \(\$ 2500\) per year. Find the total salary over a tenyear period.

Step-by-Step Solution

Answer

The total salary over the ten-year period is \$375000.

Step 1: Identify the first term, common difference and number of terms

The first term (a) of the series is the starting salary, which is \$33000. The common difference (d) is the annual raise, which is \$2500. The number of terms (n) is the number of years, which is 10.

Step 2: Use the formula of sum of an Arithmetic Series

The sum (S) of an arithmetic series can be calculated with the formula: \[S = \frac{n}{2} * (2a + (n-1)d)\]Let's substitute the given values into this formula.

Step 3: Calculate the total salary over the ten-year period

Substituting the values into the formula, we get\[S = \frac{10}{2} * (2*(33000) + (10-1)*(2500))\]Solving this, we get\[S = \$375000\]

Key Concepts

Arithmetic Series Formula

When we discuss salary growth over time using an arithmetic series, we're referring to a scenario where an initial amount (in this case, a starting salary) increases by a fixed amount periodically. The arithmetic series formula reflects this by giving us a handy tool to calculate the sum of such evenly increasing terms.

To put it into perspective, let's consider our example with the starting salary of \$33,000 that increases by \$2,500 annually. The formula for the sum of an arithmetic series is given by: \[S = \frac{n}{2} (2a + (n-1)d)\]Here, \(S\) represents the total sum of the series (the total salary after 10 years), \(n\) is the number of terms (10 years), \(a\) is the first term (the initial salary), and \(d\) is the common difference between the terms (the annual raise of \$2,500). Each piece of this equation plays a crucial role in determining the total amount over the given period.

Arithmetic Progression

The concept of an arithmetic progression is central to understanding how regular increments, like an annual raise, can be calculated over time. An arithmetic progression is a sequence of numbers in which each term after the first is found by adding a constant, called the common difference, to the previous term.

In terms of salary, if you receive \$33,000 in your first year and then get a raise of \$2,500 each following year, your salary progression will form an arithmetic sequence. This sequence would start with \(33,000, 35,500, 38,000, ...\) and so forth. Each year's salary can be found by adding the common difference \(\$2,500\) to the previous year's salary.

Summation of Arithmetic Series

To calculate the total earnings over a certain time frame with constant salary increments, we use the summation of an arithmetic series. This calculation sums all terms in the arithmetic progression, providing a total figure for the period in question.

For our given scenario of a 10-year period with a starting salary of \$33,000 and an annual raise of \$2,500, the total amount earned over the decade is determined by plugging values into our arithmetic series sum formula: \[S = \frac{10}{2} * (2*\$33,000 + (10-1)*\$2,500)\]Upon calculating, we find that the total salary earned over the ten years is \$375,000. This technique allows individuals and organizations to forecast total compensation costs or earnings over a period, ensuring financial planning is accurate and reflective of the increases over time.