Problem 15 - Algebra for College Students

Question

The 34th term of \(7,4,1,-2, \ldots\) \(-92\)

Step-by-Step Solution

Answer

The 34th term is \(-92\).

Step 1: Identify the Pattern

First, we need to determine the pattern in the sequence. We will subtract each term from the subsequent term: \(4 - 7 = -3\), \(1 - 4 = -3\), and \((-2) - 1 = -3\). This shows the sequence is an arithmetic sequence with a common difference of \(-3\).

Step 2: Write the General Formula

In an arithmetic sequence, each term can be found using the formula \(a_n = a_1 + (n-1) \times d\), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. Here, \(a_1 = 7\) and \(d = -3\).

Step 3: Substitute Values into the Formula

We are searching for the 34th term \(a_{34}\) of the sequence. Substitute \(a_1 = 7\), \(d = -3\), and \(n = 34\):\[a_{34} = 7 + (34-1) \times (-3)\].

Step 4: Perform the Calculation

Calculate the term using the formula:\[a_{34} = 7 + 33 \times (-3) = 7 - 99 = -92\].Thus, the 34th term is \(-92\).

Key Concepts

Common Difference

The concept of a common difference is pivotal in understanding arithmetic sequences. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is known as the "common difference."

If you look at any arithmetic sequence, you're essentially dealing with a linear pattern of numbers. For instance, in the given sequence of 7, 4, 1, -2, ..., the common difference is the number that you repeatedly add (or subtract) to move from one term to the next.

Here, the common difference is calculated by subtracting any term from the previous one. This is done as shown in the solution: \(4 - 7 = -3\), \(1 - 4 = -3\), and \((-2) - 1 = -3\). The result \(-3\) confirms the sequence decreases by the same amount each time.

General Term Formula

Arithmetic sequences can be understood and manipulated using their general term formula. This formula allows you to find any term in the sequence without listing all preceding terms.

The general formula for finding the \(n\)th term of an arithmetic sequence is:

• \(a_n = a_1 + (n-1) \times d\)

Here,

• \(a_n\) is the \(n\)th term you want to find,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference, and
• \(n\) is the position of the term in the sequence.

For our sequence, \(a_1 = 7\) and \(d = -3\). Using these values, you can find the 34th term through substitution. This formula makes finding any specific term quick and effective.

Sequence Patterns

Identifying sequence patterns is central to understanding sequences. In an arithmetic sequence, these patterns enable the prediction of future terms based on the defined structure.

The pattern in an arithmetic sequence is defined by its common difference. This is crucial because it makes an arithmetic sequence predictable. With each step, you either add or subtract the common difference to get to the next term. Therefore, once you have the first term and the common difference, the whole sequence can be established.

In our example, we've identified the sequence pattern as consistently decreasing by 3. Knowing this, you can determine any term in the series without directly calculating each individual term leading up to it.