Problem 57 - Algebra for College Students

Question

Solve each problem using the ideas of arithmetic sequences and series. Increasing salary. If a lab technician has a salary of \(\$ 22,000\) her first year and is due to get a \(\$ 500\) raise each year, then what will her salary be in her seventh year? (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The salary in the seventh year will be \( \$25000 \).

Step 1: Identify the First Term and Common Difference

The first term of the arithmetic sequence is the salary in the first year, which is \( a_1 = 22000 \). The common difference is the annual raise, which is \( d = 500 \).

Step 2: Use the n-th Term Formula

In an arithmetic sequence, the n-th term \(a_n\) can be found using the formula \( a_n = a_1 + (n-1)d \).

Step 3: Substitute the Known Values

Substitute the given values into the formula: \( a_7 = 22000 + (7-1) \times 500 \).

Step 4: Simplify the Expression

Calculate the expression inside the parentheses: \( 7 - 1 = 6 \). Substitute this value back into the formula: \( a_7 = 22000 + 6 \times 500 \).

Step 5: Perform the Multiplication

Multiply the common difference by 6: \( 6 \times 500 = 3000 \).

Step 6: Add the Result to the First Term

Add 3000 to the first term: \( 22000 + 3000 = 25000 \).

Step 7: Conclusion

The technician’s salary in her seventh year will be \( \$25000 \).

Key Concepts

arithmetic sequence

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This constant difference is called the 'common difference'. A common real-world example of an arithmetic sequence is a salary that increases by the same amount each year. Understanding arithmetic sequences helps us solve problems where quantities change at a steady rate over time.
An arithmetic sequence can be represented as:

• First term: \( a_1 \)
• Second term: \( a_2 = a_1 + d \)
• Third term: \( a_3 = a_1 + 2d \)
• ...and so forth.

Here, \( d \) is the common difference.

common difference

The 'common difference' in an arithmetic sequence is the amount by which each term increases (or decreases) relative to the previous term. It is denoted by \( d \).
In the context of the exercise, the lab technician’s salary increases by \$500 each year, so the common difference \( d \) is \$500.
To find the common difference in an arithmetic sequence, you can use the formula:

• \( d = a_{n} - a_{n-1} \)

where \( a_{n} \) is the n-th term and \( a_{n-1} \) is the term before it. This helps in understanding how values in a series are related.

n-th term formula

The n-th term formula for an arithmetic sequence helps us find the value of any term in the sequence without listing all the terms. The formula is:
\( a_n = a_1 + (n-1)d\)
where:

• \( a_n \) is the n-th term we want to find
• \( a_1 \) is the first term
• \( n \) is the term number
• \( d \) is the common difference

In our example, to find the lab technician’s salary in her seventh year (\( n = 7 \)), we substitute the given values into the formula:
\( a_7 = 22000 + (7-1) \times 500\)
After calculating this, we find the salary in the seventh year to be \$25,000.

salary calculation

Salary calculations using arithmetic sequences are straightforward once you understand the n-th term formula and common difference.
To find out the salary in any given year if it increases by a fixed amount annually:

• Identify the starting salary (first term \( a_1 \))
• Determine the annual raise (common difference \( d \))
• Decide the number of years (term number \( n \))
• Plug these values into the n-th term formula
• Simplify and solve the expression

For instance, if the salary starts at \$22,000 and increases by \$500 each year, after seven years, the salary will be:
\( 25000 = 22000 + 6 \times 500\)This results in a salary of \$25,000 in the seventh year. This method can be applied for various financial forecasts and educational budgeting problems.