Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, find the specified term given two terms from an arithmetic sequence. \(a_{3}=-17.1\) and \(a_{10}=-15.7\) Find \(a_{21}\).

Step-by-Step Solution

Answer

The 21st term, \(a_{21}\), is -13.5.

Step 1: Determine the Common Difference

In an arithmetic sequence, the common difference, \(d\), can be found using the formula \(d = \frac{a_{n} - a_{m}}{n - m}\). We are given \(a_{3} = -17.1\) and \(a_{10} = -15.7\). Thus, we have \(d = \frac{-15.7 - (-17.1)}{10 - 3} = \frac{1.4}{7} = 0.2\).

Step 2: Use the General Formula for the Arithmetic Sequence

The general formula for an arithmetic sequence is \(a_{n} = a_{1} + (n-1)d\). We'll use \(a_{3} = -17.1\) to find \(a_{1}\). Substitute the known values: \(-17.1 = a_{1} + (3-1) \times 0.2\). Simplifying gives \(-17.1 = a_{1} + 0.4\), thus \(a_{1} = -17.5\).

Step 3: Find the Specified Term

Now, use the general formula to find \(a_{21}\). Substitute \(n = 21\), \(a_{1} = -17.5\), and \(d = 0.2\) into \(a_{n} = a_{1} + (n-1)d\). This gives \(a_{21} = -17.5 + (21-1) \times 0.2\). Simplifying, \(a_{21} = -17.5 + 4 = -13.5\).

Key Concepts

Common Difference

In an arithmetic sequence, each term is derived by adding a fixed value, known as the common difference, to the previous term. This consistent increment is what defines the linearity of the sequence. Mathematically, if you have two terms, say \(a_m\) and \(a_n\), the common difference \(d\) can be calculated using the formula:

• \(d = \frac{a_n - a_m}{n - m}\)

Understanding the common difference allows you to predict and generate subsequent terms without guesswork. In the example where \(a_3 = -17.1\) and \(a_{10} = -15.7\), we determine the common difference \(d\) as:

• \(d = \frac{-15.7 - (-17.1)}{10 - 3} = \frac{1.4}{7} = 0.2\)

This tells us that each term in the sequence increases by 0.2 from the preceding one.

General Formula for Arithmetic Sequence

To find any term in an arithmetic sequence, we need a formula. The general formula is:

• \(a_n = a_1 + (n-1)d\)

This formula allows us to find any term \(a_n\) when we know the first term \(a_1\) and the common difference \(d\). It’s essentially a way to "count forward" in the sequence without enumerating all preceding terms.
To find \(a_1\), we rearrange the general formula using a known term. For instance, using \(a_3 = -17.1\), we solve:

• \(-17.1 = a_1 + 2 imes 0.2\)

This simplifies to:

• \(-17.1 = a_1 + 0.4\)
• \(a_1 = -17.5\)

We've found the first term, which is crucial for applying the general formula to other terms.

Finding Specified Term

Once we know the first term \(a_1\) and the common difference \(d\), we can find any specified term in the sequence with ease. Simply substitute the index of the term you want to find into the general formula.
In our example, to find \(a_{21}\), use the formula:

• \(a_{21} = a_1 + (21-1) imes d\)
• \(a_{21} = -17.5 + 20 imes 0.2\)
• \(a_{21} = -17.5 + 4\)
• \(a_{21} = -13.5\)

By applying a simple calculation, we've arrived at the term \(a_{21}\), which is \(-13.5\). This method provides a reliable step-by-step strategy to approach similar problems in arithmetic sequences.