Problem 44 - Beginning and Intermediate Algebra

Question

Determine whether each sequence is arithmetic or geometric. Then, find the general term, \(a_{m}\), of the sequence. $$\frac{3}{2}, 2, \frac{5}{2}, 3, \frac{7}{2}, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a constant difference of \(\frac{1}{2}\). The general term, \(a_m\), is given by \(a_m = \frac{m+2}{2}\).

Step 1: Determine sequence type ARITHMETIC or GEOMETRIC

In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant. Let's check if it is an arithmetic or a geometric sequence. First, we will check for an arithmetic sequence. Difference between consecutive terms: \(2-\frac{3}{2} = \frac{1}{2}\) , \(\frac{5}{2} - 2 = \frac{1}{2}\) , \(3 - \frac{5}{2} = \frac{1}{2}\) , \(\frac{7}{2} - 3 = \frac{1}{2}\) The difference between consecutive terms is constant (\(\frac{1}{2}\)), so the sequence is arithmetic.

Step 2: Finding the general term \(a_m\) of the arithmetic sequence

For an arithmetic sequence, we can find the general term using the formula: \(a_m = a_1 + (m-1)d\), where \(a_1\) is the first term, \(m\) is the term number we want to find, and \(d\) is the common difference. In our sequence, \(a_1 = \frac{3}{2}\) and \(d = \frac{1}{2}\). So, the general term of the sequence is: \(a_m = \frac{3}{2} + (m - 1)\frac{1}{2}\) Simplify: \(a_m = \frac{3+ m - 1}{2}\) Final result: \(a_m = \frac{m+2}{2}\)

Key Concepts

General Term of Sequence

In any arithmetic sequence, the pattern is determined by adding a specific number to each term to get the next term. Once you've identified this pattern, determining the general term, denoted as \( a_m \), becomes straightforward. For arithmetic sequences, the general term is given by the formula:

• \( a_m = a_1 + (m-1) \, d \)

where:

• \( a_1 \) is the first term of the sequence,
• \( m \) is the position in the sequence,
• \( d \) is the common difference between the terms.

For our sequence \( \left( \frac{3}{2}, 2, \frac{5}{2}, 3, \frac{7}{2}, \ldots \right) \), the first term \( a_1 = \frac{3}{2} \) and the common difference \( d = \frac{1}{2} \). Substitute these values into the formula to find \( a_m \):

• \( a_m = \frac{3}{2} + (m-1) \cdot \frac{1}{2} \)

Simplify to get the formula:

• \( a_m = \frac{m+2}{2} \)

This equation allows you to find any term in the sequence by just plugging in the term number you wish to find for \( m \).

Common Difference

The common difference is a key feature of an arithmetic sequence. It is the fixed number added to each term to get the next term. In other words, it's how much you "move forward" as you progress through the sequence. It's represented by \( d \) in the general term formula. To identify the common difference, \( d \), subtract any term from the previous term:

• \( d = 2 - \frac{3}{2} = \frac{1}{2} \)
• \( d = \frac{5}{2} - 2 = \frac{1}{2} \)
• \( d = 3 - \frac{5}{2} = \frac{1}{2} \)
• \( d = \frac{7}{2} - 3 = \frac{1}{2} \)

All subtractions give the same result, \( \frac{1}{2} \), confirming the sequence has a common difference and hence it is arithmetic. This common difference helps maintain the regular interval between terms, ensuring the sequence's arithmetic nature.

Sequence Type Determination

Determining whether a sequence is arithmetic or geometric is crucial before writing any formulas. For arithmetic sequences, the difference between consecutive terms is constant, while for geometric ones, the ratio between successive terms is constant.Let's figure it out using our example:

• Arithmetic Test: Check the difference between terms. We found that \( 2 - \frac{3}{2} = \frac{1}{2} \), \( \frac{5}{2} - 2 = \frac{1}{2} \), and all differences are consistent.

Because the differences are consistent, this sequence is arithmetic. For a geometric test, you'd divide each term by the previous one. However, here, the ratio is inconsistent. When the difference is consistent but not the ratio, you confirm you have an arithmetic sequence. Understanding the type of sequence you are dealing with ensures you use the appropriate formulas and properties to solve related problems accurately.