Problem 40 - Beginning and Intermediate Algebra

Question

Two terms of an arithmetic sequence are given in each problem. Find the general term of the sequence, \(a_{n}\), and find the indicated term. $$a_{4}=-10, a_{9}=0 ; a_{17}$$

Step-by-Step Solution

Answer

The general term of the arithmetic sequence is \(a_n = -16 + (n - 1)(2)\) and the 17th term, \(a_{17}\), is 16.

Step 1: Find the common difference

To find the common difference between consecutive terms in the sequence, we need to subtract the term with the lower index from the term with the higher index and then divide by the difference in the indices. Since we are given \(a_4 = -10\) and \(a_9 = 0\), we can find the common difference by using the formula: \(d = \frac{a_9 - a_4}{9 - 4}\) We can substitute the given values into this formula: \(d = \frac{0 - (-10)}{5} = \frac{10}{5} = 2\) So, the common difference, \(d\), is 2. The arithmetic sequence progresses by adding 2 to each term to find the next term.

Step 2: Write the general term for the sequence

To find the general term of the arithmetic sequence, \(a_n\), we can use the formula: \(a_n = a_1 + (n - 1)d\) We have the value for the common difference, \(d = 2\), but we need to find the first term \(a_1\). We can use one of the given terms and the value of \(d\) to determine \(a_1\). Let's use \(a_4 = -10\). Substituting these values into the general term formula, we get: \(-10 = a_1 + (4 - 1)(2)\) Now we can solve for \(a_1\): \(-10 = a_1 + 3(2)\) \(-10 = a_1 + 6\) \(a_1 = -16\) Now that we have the value for \(a_1\), we can write the general term for the sequence: \(a_n = -16 + (n - 1)(2)\)

Step 3: Find the 17th term of the sequence

To find the 17th term of the sequence, we can simply substitute \(n = 17\) into the general term formula: \(a_{17} = -16 + (17 - 1)(2)\) \(a_{17} = -16 + (16)(2)\) \(a_{17} = -16 + 32\) \(a_{17} = 16\) So, the 17th term of the sequence is 16.

Key Concepts

Common Difference

In any arithmetic sequence, the common difference is a crucial feature. It tells us how much one term changes from the one before it. This difference is constant throughout the sequence.
To find the common difference, use the formula: \(d = \frac{a_m - a_n}{m - n}\), where \(a_m\) and \(a_n\) are terms in the sequence, and \(m\) and \(n\) are their respective positions.
In the problem we are looking at, \(a_4 = -10\) and \(a_9 = 0\). Thus, the common difference \(d\) is:

• \(d = \frac{0 - (-10)}{9 - 4} = \frac{10}{5} = 2\)

This indicates that to get from one term to the next, we increase by 2 each time. The common difference is a simple yet powerful concept, as it helps us understand the structure and behavior of the whole sequence.

General Term of an Arithmetic Sequence

Once you have the common difference, you can determine the general term of an arithmetic sequence, often expressed as \(a_n\).
The general term formula is: \(a_n = a_1 + (n - 1)d\), where:

• \(a_1\) is the first term of the sequence
• \(n\) is the term number
• \(d\) is the common difference

In this exercise, we first need to find \(a_1\). We know \(a_4\) and the common difference, so:
\(-10 = a_1 + 3 \times 2\) leads to:

• \(a_1 = -16\)

Thus, the general term formula becomes:

• \(a_n = -16 + (n - 1) \times 2\)

This formula allows you to find any term in the sequence directly without having to list all the preceding terms.

Nth Term Formula

The nth term formula is simply a way of calculating any specific term in an arithmetic sequence using just one equation. This formula saves time and effort.
In our sequence, the nth term formula, given the first term \(a_1 = -16\) and the common difference \(d = 2\), is:

• \(a_n = -16 + (n - 1) \times 2\)

For example, if we wanted the 17th term, \(a_{17}\), we would substitute \(n = 17\) into the formula:

• \(a_{17} = -16 + (17 - 1) \times 2 = 16\)

This formula is extremely helpful in finding any term quickly and efficiently. It bridges the link between an arithmetic sequence's regular pattern and practical computation.