Problem 14 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 3,6,9,13, \dots $$

Step-by-Step Solution

Answer

The sequence is not arithmetic.

Step 1: Identify the Sequence

The given sequence is \( 3, 6, 9, 13, \dots \). We need to determine if this sequence is arithmetic.

Step 2: Determine the Pattern

Observe the pattern in the sequence. The difference between each consecutive term is calculated.\( 6 - 3 = 3 \), \( 9 - 6 = 3 \), \( 13 - 9 = 4 \). The differences \(3, 3,\) and \(4\) are not consistent.

Step 3: Identify Arithmetic Sequence

For a sequence to be arithmetic, the difference between consecutive terms must be the same. The differences in the sequence \( 3, 6, 9, 13, \dots \) are not consistent; hence it is not arithmetic.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the consistent value that separates each term from the next.

• If you add this value to any term, you arrive at the next term.
• For example, in an arithmetic sequence like 3, 6, 9, ..., the difference between consecutive terms is 3 (i.e., each term is increased by 3).
• This is confirmed by subtracting each term from the subsequent one, such as 6 - 3 = 3, 9 - 6 = 3, and so on.

Understanding common difference is crucial for identifying if a sequence is arithmetic. If this difference varies among the sequence, then the sequence is not arithmetic. In the case of the sequence 3, 6, 9, 13, the difference changes (last term: 13 - 9 = 4), indicating it is not an arithmetic sequence.

Sequence Pattern

To determine whether a sequence is arithmetic, it is important to examine its pattern.

• Look at how the numbers change as you move from one term to the next.
• Calculate the difference between terms to find a repeating pattern.
• In arithmetic sequences, this pattern remains consistent throughout.

If you discover any variation in the pattern, this signals that the sequence might follow a different rule. For instance, the sequence 3, 6, 9, 13 changes from consistently increasing by 3 to an increase by 4 between 9 and 13. This inconsistency indicates the pattern does not adhere to the rules defining an arithmetic sequence.

Consecutive Terms

Consecutive terms in a sequence are terms that appear one after the other without any other terms in between.

• In mathematics, especially when dealing with sequences, understanding the relationship between consecutive terms helps in recognizing patterns or types of sequences.
• For arithmetic sequences, the relationship between consecutive terms is defined by a constant difference.
• This relationship allows predictions for the next term by simply adding the common difference to the last known term.

In the explored sequence 3, 6, 9, 13, the changes between consecutive terms are not constant due to varying differences. This confirms that they do not form an arithmetic sequence, as consistent relationships are key in such sequences.