Problem 26 - College Algebra

Question

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$6,1,-4,-9, \dots$$

Step-by-Step Solution

Answer

The 20th term of the sequence is -89

Step 1: Find the Common Difference

In an arithmetic sequence, the common difference is the difference between any two successive terms. From the sequence \(6,1,-4,-9, \dots\), the common difference can be found by subtracting the first term from the second term. This gives: \(d = 1 - 6 = -5\)

Step 2: Find the First Term

The first term in the sequence is provided as \(6\). Therefore, \(a_1 = 6\).

Step 3: Create the Formula

Plug in the values of \(a_1\) and \(d\) into the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) * d\). This gives: \(a_n = 6 + (n - 1) * -5\)

Step 4: Find the 20th term

Plug \(n = 20\) into the formula: \(a_{20} = 6 + (20 - 1) * -5 = -89\)

Key Concepts

Common Difference

Understanding the common difference is crucial when dealing with arithmetic sequences. The common difference is the constant amount that each term in the sequence increases or decreases by from the previous term. To find it, we simply take any term in the sequence and subtract the one before it.

For example, considering the sequence given in the exercise, the terms are 6, 1, -4, -9, and so on. To find the common difference (\( d \)), subtract the second term (1) from the first term (6), which yields a common difference of \( d = 1 - 6 = -5 \) This negative value tells us that each subsequent term in the sequence is 5 less than the previous term.

Arithmetic Sequence Formula

The arithmetic sequence formula provides a straightforward way to find any term in the sequence without having to list out all the previous terms. The formula is \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the nth term of the sequence, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

As an exercise improvement tip, remember the '*'-sign represents multiplication in the formula. It's essential to use parentheses properly to avoid confusion. For instance, in our exercise, \( a_1 \) is 6, and the common difference \( d \) is -5, which we previously calculated. By plugging these values into our formula, we set up an expression that will correctly calculate any term in the sequence.

Sequence Term Calculation

Once you understand the arithmetic sequence formula, calculating any term in the sequence becomes a matter of plugging in the right values. For the 20th term (\( a_{20} \) ), we use \( n = 20 \).

Following our formula:

\( a_{20} = a_1 + (20 - 1)d \). Substituting the known values gives us \( a_{20} = 6 + (20 - 1) * (-5) \). Simplifying inside the parentheses first (according to the order of operations), we get \( a_{20} = 6 + 19 * (-5) \), and further simplifying leads to \( a_{20} = 6 - 95 = -89 \).

Thus, \( a_{20} = -89 \) confirms the 20th term of the sequence. This step-by-step approach is a helpful method ensuring accuracy in sequence term calculation, an essential skill in sequence analysis.