Problem 40 - Beginning and Intermediate Algebra

Question

Determine whether each sequence is arithmetic or geometric. Then, find the general term, \(a_{m}\), of the sequence. $$8,3,-2,-7,-12, \dots$$

Step-by-Step Solution

Answer

The given sequence is arithmetic with a common difference of -5. The general term for this sequence is \(a_m = 13 - 5m\).

Step 1: Determine if it is an arithmetic sequence

To check if it is an arithmetic sequence, we need to find if there is a common difference between the consecutive terms. Let's find the differences between consecutive terms: \(3 - 8 = -5\) \(-2 - 3 = -5\) \(-7 - (-2) = -5\) \(-12 - (-7) = -5\) We have a constant difference between consecutive terms, so it is an arithmetic sequence with a common difference of -5.

Step 2: Find the general term for an arithmetic sequence

Now that we know it is an arithmetic sequence, we can apply the arithmetic sequence formula to find the general term, \(a_m\): $$a_m = a_1 + (m - 1) d$$ Here, \(a_1\) is the first term of the sequence, \(d\) is the common difference, and \(m\) is the position of the term in the sequence. In our case, \(a_1 = 8\), and \(d = -5\). So, the general term is: $$a_m = 8 + (m - 1)(-5)$$ Now, we can simplify the expression: $$a_m = 8 - 5m + 5$$ $$a_m = 13 - 5m$$ The general term for the given arithmetic sequence is: $$a_m = 13 - 5m$$

Key Concepts

Common Difference

In an arithmetic sequence, the key feature is the 'common difference' between consecutive terms. The common difference remains consistent throughout the entire sequence. This regularity makes arithmetic sequences predictable and easy to work with. To determine the common difference, simply subtract any term from the following term in the sequence.
For example, given the sequence:

• 3 - 8 = -5
• -2 - 3 = -5
• -7 - (-2) = -5
• -12 - (-7) = -5

In this sequence, each pair of terms has a difference of -5, confirming that the sequence is arithmetic. Once found, the common difference helps in constructing the broader sequence and deriving its general term.

General Term

The 'general term' in an arithmetic sequence is crucial because it gives a formula for finding any term in the sequence. Openness to understanding this formula transforms a pattern into an analytical tool. The general term is denoted as \(a_m\), where \(m\) is the position of a term in the sequence.
To find the general term for an arithmetic sequence, we employ the formula:\[a_m = a_1 + (m - 1) \cdot d\]Here:

• \(a_1\) is the first term of the sequence
• \(d\) is the common difference
• \(m\) is the position of the term

In our example with the sequence 8, 3, -2, -7, -12, ... where \(a_1 = 8\) and \(d = -5\), the general formula becomes \(a_m = 8 + (m - 1)(-5)\), which simplifies to \(a_m = 13 - 5m\). This formula allows you to find any term in the sequence just by substituting \(m\) with the term number you wish to find.

Sequence Formula

The 'sequence formula' is a foundation in mathematics for describing sequences, like the arithmetic sequence mentioned here. For arithmetic sequences, this formula is a streamlined approach for determining any term without listing all preceding terms.
The sequence formula as described by:\[a_m = a_1 + (m - 1) \cdot d\] Is particularly effective because:

• It abstracts the arithmetic pattern into a simple algebraic expression.
• It can be used to find any term quickly and efficiently.
• It provides insight into the structure and behavior of the sequence.

For our example sequence starting with 8 and a common difference of -5, the sequence formula simplifies into \(a_m = 13 - 5m\). Using this formula, you bypass iterative addition and can directly compute the value of any sequence term using its position \(m\). This makes strategic uses in settings like computer algorithms or predictive models where time and efficiency are priorities.