Arithmetic Sequence nth-Term Formula with Examples

Count the increments before writing the formula

An arithmetic sequence adds the same difference d at every step. Starting from a1, the second term is a1 + d, the third is a1 + 2d, and the nth is a1 + (n − 1)d. There are n − 1 increments because the starting value already occupies position one. Using nd instead would move one step too far and return the next term. Substitute n = 1 as a quick check: the increment should vanish and leave a1.

Find a distant term directly

For a1 = 7 and d = 4, the first terms are 7, 11, 15 and 19. The twentieth term is 7 + 19×4 = 83. Enter a = 7, d = 4 and n = 20 in the calculator. Its finite-sum output is 20(7 + 83)/2 = 900, which answers a different question. Reporting 900 when asked for the twentieth term would confuse the total of twenty values with the single value at position twenty.

Handle a negative common difference

For a1 = 30 and d = −3, the eighth term is 30 + 7(−3) = 9. The sequence decreases, but the position count remains positive. Continuing the rule can eventually produce zero or negative terms, which is mathematically valid unless the application imposes a separate restriction. If the values represent a finite inventory, for example, the model may stop before a negative amount becomes meaningful. That application boundary is distinct from the arithmetic formula itself.

Recover the position of a target term

To find where 83 occurs in the first example, solve 83 = 7 + (n − 1)4. Subtracting 7 and dividing by 4 gives n − 1 = 19, so n = 20. A target of 84 would give a noninteger position and therefore is not a term in this discrete sequence. The local form expects n as an input; solve the position equation separately and then use the tool to verify the resulting integer candidate. A constant sequence with d = 0 needs separate interpretation when solving for a position.

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

Find an Arithmetic Sequence’s Common Difference

Find a common difference from adjacent or separated terms, keep subtraction order consistent, and reject a list whose differences change.

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.

Find the nth Term of a Geometric Sequence

Use a1 times r to the power n minus 1, handle negative and fractional ratios, and distinguish a geometric term from its finite sum.