Problem 66 - Algebra 2

Question

Explain why the sequence 4, 5, 7, 10, 14, ... is not arithmetic.

Step-by-Step Solution

Answer

The sequence isn't arithmetic because differences (1, 2, 3, 4) between terms are not constant.

Step 1: Define an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is the result of adding a constant to the previous term. This constant is known as the "common difference." For an arithmetic sequence, the difference between consecutive terms should be the same for all pairs of consecutive terms.

Step 2: Identify Consecutive Differences

To determine if the given sequence is arithmetic, calculate the difference between each pair of consecutive terms. For the sequence 4, 5, 7, 10, 14:- Difference between second term (5) and first term (4): \[ 5 - 4 = 1 \]- Difference between third term (7) and second term (5): \[ 7 - 5 = 2 \]- Difference between fourth term (10) and third term (7): \[ 10 - 7 = 3 \]- Difference between fifth term (14) and fourth term (10): \[ 14 - 10 = 4 \]

Step 3: Analyze the Differences

Examine the differences calculated: - The differences are 1, 2, 3, and 4. These differences are not constant; they vary between the pairs of consecutive terms. This inconsistency indicates that the sequence does not have a common difference and, therefore, is not arithmetic.

Key Concepts

Common Difference

In an arithmetic sequence, a crucial concept is the "common difference." This is the fixed number that you add to each term to get the next term in the sequence. Imagine it like the step size in a pattern of numbers.
For example, let's consider the simple sequence 3, 5, 7, 9, 11. Here, the common difference is 2. You add 2 to 3 to get 5, then add 2 to 5 to get 7, and so on.
To find the common difference in any sequence, subtract any term from the next term.

• If the result is the same for all consecutive pairs, you have found the common difference.
• This uniform step is what makes the sequence arithmetic.

If the differences are not equal, the sequence is not arithmetic, as seen in the sequence 4, 5, 7, 10, 14 where differences vary at each step.

Consecutive Terms

"Consecutive terms" refer to numbers that follow one another in the sequence. To understand a sequence, it's essential to look at these pairs of numbers and examine how they relate to one another.
In the context of arithmetic sequences, consecutive terms are especially important. They help you identify the common difference through subtraction between them.
For instance, in the sequence 1, 4, 7, 10:

• The consecutive terms 1 and 4 have a difference of 3.
• Similarly, 4 and 7 also have a difference of 3, affirming it's an arithmetic sequence.

In contrast, if you consider consecutive terms in the series 4, 5, 7, 10, 14:

• The differences are 1, 2, 3, and 4.
• Because these differences are not the same, consecutive terms alert us that this is not an arithmetic sequence.

Determine if a Sequence is Arithmetic

To determine if a sequence is arithmetic, you need to check whether there is a consistent pattern or rule throughout the sequence. The process involves observing the differences between consecutive terms.
Start by selecting the first few terms of the sequence, and subtract each term from the one that follows it.
For example, with the sequence 3, 6, 9, 12:

• Difference between second and first term: \(6 - 3 = 3\)
• Difference between third and second term: \(9 - 6 = 3\)
• Difference between fourth and third term: \(12 - 9 = 3\)

As all these differences are the same, the sequence is arithmetic.
However, in the sequence 4, 5, 7, 10, 14, calculations reveal differences of 1, 2, 3, and 4 respectively. These differing values indicate the sequence lacks a common difference, classifying it as non-arithmetic.