Find an Arithmetic Sequence’s Common Difference

Subtract a term from the next one

For the list 18, 13, 8, 3, the consecutive differences are 13 − 18 = −5, 8 − 13 = −5 and 3 − 8 = −5. The common difference is −5. Use later minus earlier in every subtraction; reversing just one pair changes its sign and can make a consistent list appear inconsistent. The word difference here means a signed change, not the absolute distance between values.

Use separated terms and their positions

If a3 = 11 and a9 = 35 in an arithmetic sequence, there are six equal steps between the positions. Thus d = (35 − 11)/(9 − 3) = 24/6 = 4. There are seven listed positions from three through nine inclusive, but only six intervals between them. Counting the positions instead of the intervals gives the wrong divisor. This is the same reason the nth-term formula uses n − 1 increments from the first term.

Recover the first term and verify the data

With a3 = 11 and d = 4, the first term is a1 = 11 − 2×4 = 3. The sequence begins 3, 7, 11, 15 and continues to a9 = 35. Enter a = 3, d = 4 and n = 9 to check the ninth-term result. Substituting into both supplied positions is useful: matching only one known term would not reveal a mistaken difference paired with a compensating starting value.

Identify a list that is not arithmetic

For 2, 5, 9, 14, consecutive differences are 3, 4 and 5, so there is no common first difference. Averaging those differences produces 4, but does not turn the original list into an arithmetic sequence. The resulting model would miss some supplied terms. For measured data with small discrepancies, an approximate trend might be useful, but it needs an explicit fitting method and error discussion rather than an exact arithmetic-sequence claim. For exact fractional data such as 1/2, 3/4 and 1, the common difference is 1/4; fractions need not be rounded to identify the arithmetic pattern.

State the first index and the starting value

A sequence assigns a value to each allowed position. In the arithmetic and geometric formulas here, the first position is n = 1 and its value is a1. The local sequence forms follow that convention. If a problem instead begins at index zero, its initial value is a0 and the exponent or increment count shifts accordingly. Write the index beside the starting value before substituting numbers. An otherwise correct formula can be one term out if the starting convention changes between a worksheet, a program and the calculator.

Distinguish a term from a sum

The nth term is the value at one position, while the sum of the first n terms adds all values from the starting position through n. These can be very different numbers. Read the output label before copying an answer. The local sequence tools show both a term and a finite sum, plus a preview of initial terms. A short preview does not mean the calculation stopped at the last displayed preview value. Use a small example that can also be added or generated by hand when checking which output the question requests.

Check that the proposed pattern is justified

A short list can fit more than one mathematical rule. Constant consecutive differences support an arithmetic model, while constant ratios between nonzero consecutive terms support a geometric model. A problem should state the intended pattern or provide enough context to choose it. Do not force an arbitrary measurement series into one of these models merely because a calculator accepts its first two numbers. Preserve fractions or adequate intermediate precision when deriving a difference or ratio, then round only the final requested result. Negative terms and decreasing patterns can be valid; the position n still needs to be a whole number.

Related guides

Arithmetic Sequence nth-Term Formula with Examples

Use a1 plus (n minus 1)d to find a distant arithmetic-sequence term, explain the index offset, and recover a missing position.

Calculate Arithmetic Terms and Their Finite Sum

Generate an arithmetic sequence, calculate its nth term and finite sum, and verify the result by pairing the first and last terms.

Identify a Geometric Sequence and Find Its Sum

Check consecutive ratios, construct a geometric rule, calculate a finite sum, and distinguish finite sums from a convergent infinite series.