Arithmetic Sequence Calculator

Some patterns are friendly: each term is just the previous one plus the same fixed step. The arithmetic sequence calculator works with exactly those patterns, finding any term in the sequence and the sum of its first stretch of terms without listing them all out by hand. It earns its keep on homework about sequences and series, and on real problems with the same shape — savings plans that grow by a fixed amount, seating rows, stacked objects, training schedules.

Quick Answer

n-th term: aₙ = a₁ + (n − 1) × d, and sum of the first n terms: Sₙ = (n ÷ 2) × (a₁ + aₙ). Here a₁ is the first term and d is the common difference. For the sequence starting at 4 with difference 3, the 10th term is 4 + 9 × 3 = 31.

Anatomy of an Arithmetic Sequence

A sequence earns the label "arithmetic" when the gap between consecutive terms never changes. That constant gap, the common difference d, is the engine of everything: subtract any term from the next and you get d, whether you test the first pair or the fiftieth. Differences can be positive (sequence climbs), negative (it descends, like 20, 17, 14, … with d = −3), or zero (a constant sequence, the boring but valid edge case). Fractions and decimals work too — 1.5, 2.25, 3.0, … steps by 0.75.

The n-th term formula follows from counting steps. To reach the second term you take one step of size d from a₁; to reach the third, two steps. Reaching the n-th term takes n − 1 steps, hence aₙ = a₁ + (n − 1)d. That (n − 1) instead of n is the most-tested detail in the whole topic. The sum formula has a prettier origin, usually credited to the young Gauss: write the series forwards and backwards, add the two rows, and every column totals a₁ + aₙ. With n columns, the doubled sum is n × (a₁ + aₙ), so the sum itself is half of that. A bonus consequence: the average of an arithmetic sequence's terms equals the average of its first and last terms — the middle term, when n is odd, is the mean of the whole run.

Do not confuse this family with geometric sequences, where each term is the previous one multiplied by a fixed ratio. Addition versus multiplication sounds like a small distinction; in behaviour it is the difference between a staircase and a rocket.

How to Use the Arithmetic Sequence Calculator

1. Read the first term off your sequence and enter it as a₁.
2. Find the common difference by subtracting any term from the one after it, and enter it as d.
3. Double-check d with a second pair of terms — if the gaps differ, the sequence is not arithmetic and this tool does not apply.
4. Enter the term position n you want (for the n-th term) or the number of terms to add (for the sum).
5. Read the term value and the sum, and spot-check the first few terms by hand against the pattern.

First Term 4, Difference 3, Ten Terms Deep

The sequence runs 4, 7, 10, 13, 16, 19, 22, 25, 28, 31. The 10th term from the formula: a₁₀ = 4 + (10 − 1) × 3 = 4 + 27 = 31, matching the list. The fifth term is a₅ = 4 + 4 × 3 = 16, the middle of the list. Now the sum of all ten terms using Gauss's pairing: S₁₀ = (10 ÷ 2) × (4 + 31) = 5 × 35 = 175. Add the listed terms directly as a check — the running total ends at 175, confirmed. Even the shortcut for five terms agrees: S₅ = (5 ÷ 2) × (4 + 16) = 50, and 4 + 7 + 10 + 13 + 16 = 50.

Notice how the middle term 16 relates to the sum: 175 ÷ 10 = 17.5 is the average of all ten terms, and (4 + 31) ÷ 2 = 17.5 — the first-last average, exactly as promised.

The Off-by-One Error and Other Sequence Snares

Nearly every wrong answer in this topic comes from writing a₁ + n × d instead of a₁ + (n − 1) × d. The first term required zero steps to reach, so the step count always trails the position by one. The second snare is a difference computed backwards — subtracting the larger term from the smaller regardless of order flips the sign of d and sends a climbing sequence downhill. Always subtract in sequence order: later term minus earlier term.

A third trap is assuming arithmetic structure where none exists. The sequence 2, 6, 12, 20 has growing gaps (4, 6, 8) and is not arithmetic; feeding its first gap into this calculator produces confident nonsense. Verify two gaps before trusting the model. And when a word problem hides the sequence — payments rising by a fixed amount yearly, bricks per row in a pyramid — extract a₁ and d explicitly before calculating; most errors happen in the translation from words to numbers, not in the arithmetic.

A Savings Plan Is an Arithmetic Sequence

Suppose you save $50 in January and increase the deposit by $10 every month — $50, $60, $70, and so on. That is an arithmetic sequence with a₁ = 50 and d = 10. December's deposit is a₁₂ = 50 + 11 × 10 = $160, and the year-end total is S₁₂ = (12 ÷ 2) × (50 + 160) = 6 × 210 = $1,260. The same model prices anything that steps evenly: rent rising by a fixed amount each year, production quotas climbing by a set number of units per week, or a runner adding the same distance to every long run. In each case the two formulas split the question cleanly — the term formula tells you the size of any single step far in the future, and the sum formula tells you the accumulated total when you get there, without listing a single intermediate month.

Sequences Lead Onward

Mastering arithmetic sequences sets up the rest of series work. The summation calculator totals arbitrary lists and series, the average calculator connects to the first-and-last-term mean property you saw above, and the broader math calculators collection covers the neighbouring topics, from factorials to standard deviation.