Quick answer
Subtract consecutive terms: d = a₂ - a₁. Repeat for multiple pairs to confirm the pattern is arithmetic.
Formula
- d = a₂ - a₁
- d = aₙ - aₙ₋₁ for consecutive terms
Introduction
Common difference is the heartbeat of an arithmetic sequence. If d is wrong, every later term and every series sum built on those terms will drift.
After you stabilize d, apply the nth term formula to reach any position.
See worked examples for lists where d is positive, negative, and fractional.
Definition and identification
d is the fixed change between neighbors. Positive d steps up; negative d steps down; fractional d steps by decimals.
Identification begins with subtraction, not with guessing from a graph. Graphs help later; subtraction proves constancy.
If three consecutive differences are 4, 4, and 4, the list is arithmetic with d = 4. If they differ, look for another model.
Formulas and pattern checks
- d = a₂ - a₁
- d = (aₙ - a₁) / (n - 1) when you know distant terms
Use the distant-term version when a problem gives a₁ and a₁₀ but not intermediate values. Solve for d algebraically, then verify on neighbors if you construct intermediates.
Increasing vs decreasing is only about sign. The absolute size of d tells you how fast the list moves.
Error checking: re-subtract two neighbors after you finish a long list. One late mistake in addition can fake a non-constant pattern.
Step-by-step guide
- Pick a consistent order. Compute later minus earlier every time.
- Subtract at least two pairs. One pair can mislead if a typo appears.
- Record sign. Negative d is valid and common in temperature or debt stories.
- Apply the nth term formula. Use verified d with a₁ to generate more terms.
Increasing and decreasing samples
List 5, 9, 13: differences are 4, so d = 4.
List 8, 5, 2, -1: differences are -3, so d = -3.
If a problem states the list is arithmetic but differences vary, revisit whether you copied terms correctly before changing models.