What is the definition of a common difference?

The common difference is the constant value added to each term in an arithmetic sequence to generate the next term. In other words, if you subtract any term from the term that immediately follows it in the sequence, you will always get the same value, and that value is the common difference.

To illustrate, consider the following:

• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
• Common Difference (d): The constant difference between consecutive terms in an arithmetic sequence.

How to Find the Common Difference:

To find the common difference (d) in an arithmetic sequence, simply subtract any term from the term that follows it:

d = a_n+1 - a_n

Where:

• a_n+1 is the (n+1)^th term in the sequence
• a_n is the n^th term in the sequence

Examples:

1. Sequence: 2, 5, 8, 11, 14,...

   • 5 - 2 = 3
   • 8 - 5 = 3
   • 11 - 8 = 3
   • 14 - 11 = 3
   • The common difference is 3.

2. Sequence: 10, 7, 4, 1, -2,...

   • 7 - 10 = -3
   • 4 - 7 = -3
   • 1 - 4 = -3
   • -2 - 1 = -3
   • The common difference is -3.

Key Points:

• A common difference can be positive, negative, or zero.
• If a sequence has a common difference, it is an arithmetic sequence.
• The common difference is crucial for determining the general term of an arithmetic sequence.

Therefore, the common difference is the foundation for identifying and working with arithmetic sequences. It represents the consistent rate of change between consecutive terms within the sequence.