Problem 31 - Algebra 2

Question

Is the sequence arithmetic? If so, identify the common difference. $$ 34,51,68, \ldots $$

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic and the common difference is 17.

Step 1: Is the sequence arithmetic?

To find out whether the sequence is arithmetic or not, subtract the first term from the second term, and then subtract the second term from the third term. Here, \(51 - 34 = 17\) and \(68 - 51 = 17\). Since the difference is the same, we conclude that this sequence is arithmetic.

Step 2: Identify the common difference

In an arithmetic sequence, the common difference is the constant difference between consecutive terms, which we already calculated in previous step as \(17\). Therefore, the common difference in this sequence is \(17\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key concept. It is the fixed amount that each term increases or decreases to get to the next term. This value remains consistent throughout the sequence.

To find the common difference, you take any two consecutive terms in the sequence and subtract the first from the second. For example, in the sequence 34, 51, 68, the common difference is calculated as follows:

• Subtract the first term from the second: \(51 - 34 = 17\).
• Subtract the second term from the third: \(68 - 51 = 17\).

Since the result is the same in both cases, we confirm that 17 is the common difference for this sequence.

This value indicates how rapidly the sequence is increasing. Whether calculating forward or backward in the sequence, this common difference helps understand the sequence's behavior.

Consecutive Terms

Consecutive terms in an arithmetic sequence are terms that directly follow one another. For instance, in the sequence 34, 51, 68, each number directly follows or precedes another.

The importance of consecutive terms lies in identifying the sequence's pattern. They allow us to determine if a sequence is arithmetic by checking if their differences remain constant. If you subtract the first consecutive term from the second and get the same difference repeatedly, you are likely dealing with an arithmetic sequence.

This sequence property simplifies prediction of future terms. If terms remain consistent with the common difference, finding any term in the sequence becomes straightforward. You just need the starting point and the common difference. Understanding consecutive terms is critical to recognizing and working with arithmetic sequences effectively.

Difference Calculation

The process of calculating the difference between terms in a sequence is fundamental to determining if the sequence is arithmetic. This involves simple subtraction between any two consecutive terms.

The difference calculation is performed by taking the second term and subtracting the first, then repeating this between other pairs of terms.

• Example: For the terms 34 and 51, the difference is \(51 - 34 = 17\).
• For the terms 51 and 68, the difference is \(68 - 51 = 17\).

If all these differences are equal, as they are in this example, the sequence is confirmed as arithmetic, and we identify this repeated value as the common difference. This methodical approach gives clarity and helps avoid errors, making it a foolproof technique for analyzing arithmetic sequences.