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Common Difference — Definition, Formula & Examples

The common difference is the fixed amount you add (or subtract) to get from one term to the next in an arithmetic sequence. It stays the same throughout the entire sequence.

For an arithmetic sequence a1,a2,a3,…a_1, a_2, a_3, \ldots, the common difference dd is the constant d=an+1−and = a_{n+1} - a_n for all positive integers nn.

Key Formula

d=an+1−and = a_{n+1} - a_n
Where:
  • dd = The common difference
  • ana_n = The current term in the sequence
  • an+1a_{n+1} = The next term in the sequence

How It Works

To find the common difference, subtract any term from the term that follows it. If you get the same result for every pair of consecutive terms, the sequence is arithmetic and that result is dd. A positive dd means the sequence increases, a negative dd means it decreases, and d=0d = 0 produces a constant sequence. Once you know dd and the first term a1a_1, you can generate the entire sequence or jump directly to any term using an=a1+(n−1)da_n = a_1 + (n-1)d.

Worked Example

Problem: Find the common difference of the arithmetic sequence 7, 13, 19, 25, 31, … and then find the 10th term.
Find d: Subtract the first term from the second term.
d=13−7=6d = 13 - 7 = 6
Verify: Check another pair to confirm the difference is constant.
19−13=6✓19 - 13 = 6 \quad \checkmark
Find the 10th term: Use the nth-term formula with a1=7a_1 = 7, d=6d = 6, and n=10n = 10.
a10=7+(10−1)(6)=7+54=61a_{10} = 7 + (10-1)(6) = 7 + 54 = 61
Answer: The common difference is d=6d = 6, and the 10th term is 61.

Why It Matters

The common difference appears throughout Algebra 2 and Precalculus whenever you model situations with steady, constant change — monthly savings deposits, evenly spaced seating rows, or uniform dose increases. Recognizing dd quickly also lets you set up arithmetic series to find totals without adding every term individually.

Common Mistakes

Mistake: Confusing common difference with common ratio.
Correction: The common difference is found by subtracting consecutive terms (an+1−ana_{n+1} - a_n). The common ratio is found by dividing them (an+1/ana_{n+1} / a_n) and belongs to geometric sequences, not arithmetic ones.