How to Find n in an Arithmetic Series: Step-by-Step

To find n in an arithmetic series, use the nth term formula aₙ = a + (n − 1)d when the first term, common difference, and last term are known, or the sum formula Sₙ = n/2[2a + (n − 1)d] when the sum is given instead. In the first case, n = (aₙ − a)/d + 1; in the second, you solve a quadratic for n. Everything depends on identifying which three values the problem hands you, then substituting into the matching formula.

Identify the known values

An arithmetic sequence is an ordered list where consecutive terms differ by a constant, and an arithmetic series is the sum of those terms. Five quantities matter:

• a — the first term.
• d — the common difference (negative if the sequence decreases).
• aₙ — the nth term, sometimes called the last term when the sequence stops.
• n — the position of that term.
• Sₙ — the sum of the first n terms.

Read for two signals: whether the problem mentions a total or sum, and whether it gives a final term. “The sum of the first n terms is 155” points to the sum formula. “The last term is 98” or “how many terms until the value reaches 63” points to the nth term formula. If you know a, d, and aₙ, you solve for n directly. If you know a, d, and Sₙ, you set up a quadratic.

Watch the sign of d. In the sequence 50, 45, 40, …, d is −5, not 5 — a common early slip.

Use the nth term formula

The nth term formula is aₙ = a + (n − 1)d. It says: start at a, then add d a total of (n − 1) times to reach the nth term. Rearranging:

n = (aₙ − a)/d + 1

The “+1” appears because moving from the first term to the nth term takes exactly n − 1 steps, so the number of terms is the number of steps plus one. Forgetting it is the most frequent mistake in these problems.

Example. An arithmetic sequence begins 11, 14, 17, … and its last term is 98. How many terms are there? Here a = 11, d = 3, aₙ = 98. Apply n = (aₙ − a)/d + 1 = (98 − 11)/3 + 1 = 87/3 + 1 = 29 + 1 = 30. The sequence has 30 terms. Notice that 87/3 alone gives 29 — the correct count is 30. Khan Academy’s arithmetic sequence lessons walk through this reasoning with step diagrams, which helps the “+1” stick.

Use the sum formula

When the problem gives the total instead of the final term, reach for Sₙ = n/2[2a + (n − 1)d]. Substitute a, d, and Sₙ, then solve for n. Because the unknown n appears in two places — once multiplying and once inside the bracket — you usually end up with a quadratic equation.

Multiply both sides by 2 to clear the fraction, expand, and gather everything on one side so it reads An² + Bn + C = 0. Then factor, or apply the quadratic formula:

n = [−B ± √(B² − 4AC)] / 2A

Only a positive whole-number answer makes sense, since n counts terms. Discard any negative or non-integer root. Rounding is not appropriate here — if the quadratic doesn’t produce a clean integer, re-check your setup and arithmetic. Wolfram MathWorld’s arithmetic progression entry lists both formulas together, useful for double-checking before you commit to the algebra.

Worked example: the sum case

Find how many terms of the series 5 + 9 + 13 + … add up to 510. Here a = 5, d = 4, Sₙ = 510. Substitute: 510 = n/2[10 + 4(n − 1)] = n/2(4n + 6). Multiply by 2: 1,020 = 4n² + 6n, so 4n² + 6n − 1,020 = 0, which simplifies to 2n² + 3n − 510 = 0. Factor: (2n + 34)(n − 15) = 0 gives n = 15 (rejecting n = −17). So 15 terms of the series sum to 510.

Common pitfalls:

• Forgetting the “+1” in the nth term rearrangement.
• Using d with the wrong sign in a decreasing sequence.
• Keeping a negative or non-integer root of the quadratic.
• Confusing the nth term aₙ with the sum Sₙ when reading the prompt.
• Plugging the given last term into the sum formula as if it were n.

Key Takeaways

• Use aₙ = a + (n − 1)d, rearranged to n = (aₙ − a)/d + 1, when you know the first term, common difference, and final term.
• Use Sₙ = n/2[2a + (n − 1)d] when the sum is given; expect to solve a quadratic in n.
• Accept only a positive whole-number value for n — discard negative and non-integer roots.
• Mind the sign of d and the “+1” step count; both cause the most frequent errors.
• Verify by substituting your n back into the original formula or by adding terms directly.