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How to Find the Common Difference of an Arithmetic Sequence?

Published in Uncategorized 2 mins read

The common difference in an arithmetic sequence is found by subtracting any term from the term that follows it. This constant difference is what defines an arithmetic sequence.

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between consecutive terms remains constant. This constant difference is what we call the common difference. For example, in the sequence 2, 5, 8, 11, 14..., the common difference is 3 (5-2 = 3, 8-5 = 3, and so on).

Calculating the Common Difference

There are several ways to calculate the common difference (often denoted as 'd'):

  • Method 1: Subtracting Consecutive Terms: The simplest method is to subtract any term from the term immediately following it.

    • Formula: d = a<sub>n</sub> - a<sub>n-1</sub> where a<sub>n</sub> represents the nth term and a<sub>n-1</sub> represents the term before it.

    • Example: In the sequence 7, 10, 13, 16..., d = 10 - 7 = 3. You can verify this by subtracting any consecutive pair: 13 - 10 = 3, 16 - 13 = 3.

  • Method 2: Using Two Terms and their Positions: If you know two terms and their positions within the sequence, you can also find the common difference.

    • Formula: d = (a<sub>p</sub> - a<sub>q</sub>) / (p - q) where a<sub>p</sub> is the term at position p, and a<sub>q</sub> is the term at position q.

    • Example: Let's say the 3rd term (a3) is 13 and the 7th term (a7) is 25. Then: d = (25 - 13) / (7 - 3) = 12 / 4 = 3.

  • Method 3: First and Second Terms: The most straightforward approach involves simply subtracting the first term from the second term. This is because the common difference is consistently the same between any two consecutive terms.

    • Example: In the sequence 5, 8, 11, 14..., d = 8 - 5 = 3.

Practical Insights:

  • The common difference can be positive, negative, or zero.
  • A positive common difference indicates an increasing sequence, while a negative common difference indicates a decreasing sequence. A common difference of zero means the sequence is constant.
  • The common difference is a crucial component in various arithmetic sequence formulas, including finding the nth term and the sum of the first n terms.

[Arithmetic Sequences]

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