Problem 60 - College Algebra

Question

Use a system of two equations in two variables, \(a_{1}\) and \(d,\) to solve Exercises \(59-60\) Write a formula for the general term (the \(n\) th term) of the arithmetic sequence whose third term, \(a_{3},\) is 7 and whose eighth term, \(a_{8},\) is 17

Step-by-Step Solution

Answer

The formula for the \(n^{th}\) term of the arithmetic sequence is \(a_n = 3 + (n-1) * 2\).

Step 1: Formulate the Equations

An arithmetic sequence can be represented as \(a_n = a_1 + (n-1) *d\). Thus, the given conditions result in two equations. The first is \(a_3 = a_1 + 2d = 7\), and the second is \(a_8 = a_1 + 7d = 17\).

Step 2: Solve the System of Equations

Subtract the first equation from the second to eliminate the \(a_1\) term. This gives \(5d = 10\), which can be solved to find \(d = 2\). Substituting \(d = 2\) into the first equation yields \(a_1 = 7 - 2*2 = 3\).

Step 3: Create the General Formula

Armed with the values for the first term and the common difference, we now form the formula for the \(n^{th}\) term of the arithmetic sequence. This formula is \(a_n = a_1 + (n-1) *d\), which, after substituting \(a_1 = 3\) and \(d = 2\), becomes \(a_n = 3 + (n-1) * 2\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same variables. To solve problems involving arithmetic sequences, we can often use a system of equations.
In our example, we are given two pieces of information about an arithmetic sequence:

• The third term, \(a_3\), is 7
• The eighth term, \(a_8\), is 17

From this, we establish two equations based on the general formula of an arithmetic sequence, which we'll discuss next. Solving these equations together helps us find unknown values like the initial term or the common difference.

General Term Formula

The general term formula for an arithmetic sequence allows you to calculate any term in the sequence. This formula is expressed as:

• \(a_n = a_1 + (n-1) \times d\)

Here, \(a_n\) represents the \(n^{th}\) term, \(a_1\) is the first term, and \(d\) is the common difference between terms.
To find the general term formula, we first had to determine the first term and the common difference using the equations derived from our given terms. Once we had \(a_1 = 3\) and \(d = 2\), it was straightforward to write the formula as \(a_n = 3 + (n-1) \times 2\). This formula allows us to find any term in this particular arithmetic sequence.

Common Difference

In an arithmetic sequence, the common difference, \(d\), is the constant amount added to each term to get the next. It is a crucial part of understanding how the sequence progresses.
To calculate \(d\), we took the two equations from our problem:

• \(a_3 = a_1 + 2d = 7\)
• \(a_8 = a_1 + 7d = 17\)

By subtracting these equations, we eliminated \(a_1\), leaving \(5d = 10\). Solving for \(d\) gave us a common difference of 2. This tells us that each term is 2 more than the previous term.

Sequence Terms

Sequence terms are the elements of the sequence that are connected through this common difference. Arithmetic sequence terms are calculated using the general term formula we derived earlier.
Let's explore some terms of our specific sequence with \(a_1 = 3\) and \(d = 2\):

• \(a_1 = 3\)
• \(a_2 = 3 + 1 \times 2 = 5\)
• \(a_3 = 3 + 2 \times 2 = 7\)
• \(a_4 = 3 + 3 \times 2 = 9\)

You can continue this pattern to find any other term in this sequence. Each term builds upon the previous one by adding the common difference. Understanding sequence terms is vital as it gives us the ability to predict and calculate future or unknown values within a sequence.