Arithmetic Sequence
An arithmetic sequence increases by a fixed common difference d. This calculator finds the nth term with aₙ = a₁ + (n − 1)d and the sum of the first n terms with Sₙ = n/2·(a₁ + aₙ).
The nth term and the running total
An arithmetic sequence advances by the same amount at every step. Seat numbering along a row, the rungs of a ladder at fixed spacing, simple interest accruing at a flat annual amount — all of these step by a constant, and that constant is the common difference d.
Given the first term, the common difference and a term number, this calculator returns two things: the value of that term, and the sum of every term up to and including it.
The nth term formula removes the need to iterate. Finding the two-hundredth term by repeated addition means 199 additions and 199 chances to slip; the closed form gets there in one multiplication.
The partial sum is the more useful of the two in practice. It answers questions like the total of the first fifty payments, or the number of seats in a wedge-shaped auditorium, which would otherwise require adding a long list of numbers by hand.
How to use this calculator
- Enter the first term a₁ The value the sequence starts from. Any real number is accepted, including negatives and decimals.
- Enter the common difference d The fixed step between consecutive terms, found by subtracting any term from the one after it. A negative d gives a decreasing sequence, and d = 0 gives a constant one.
- Enter the term number n Which term you want, counting the first as n = 1. This must be a positive whole number: the solver rejects zero, negatives and decimals with an explicit message, because a sequence has no half-th term.
- Read both outputs The answer line gives the nth term and the partial sum together. The steps above it show the substitution for each, so a disagreement with your own working can be traced to the exact line.
The formula, and where it comes from
aₙ = a₁ + (n − 1)d Sₙ = (n / 2)·(a₁ + aₙ)
The nth term formula counts steps rather than terms. Reaching the fifth term from the first takes four steps, not five, which is why the multiplier is n − 1 and not n. Nearly every off-by-one error with arithmetic sequences comes from that distinction.
The sum formula is the pairing argument attributed to the young Gauss: write the series forwards and backwards, add the two rows term by term, and every column totals a₁ + aₙ. There are n such columns and the result is twice the sum, so Sₙ = (n/2)(a₁ + aₙ) — the number of terms times the average of the first and last.
What each input means
- a₁ First term — form field “First term a₁”
- The value at n = 1. The calculator indexes from one, not zero, so a₁ is genuinely the first term and not an offset before the sequence starts.
- d Common difference — form field “Common difference d”
- The constant added at each step. If the difference between consecutive terms is not the same everywhere, the sequence is not arithmetic and neither formula applies.
- n Term number — form field “Term number n”
- A positive integer. It is both the index of the term reported and the number of terms included in the sum, which is why the two outputs always refer to the same point in the sequence.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
An increasing sequence
Take the sequence starting at 3 and rising by 5 each step: 3, 8, 13, 18, and so on. What is the tenth term, and what do the first ten terms add to?
Inputs First term a₁ = 3, Common difference d = 5, Term number n = 10
- Given a₁ = 3, common difference d = 5, n = 10
- nth term aₙ = a₁ + (n−1)d = 3 + 9·5 = 48
- Sum of n terms Sₙ = n/2·(a₁ + aₙ) = 10/2·(3 + 48) = 255
Result a₁₀ = 48, Sₙ = 255
The multiplier in the nth-term line is 9, not 10 — nine steps separate the first term from the tenth. Reading that line is the quickest way to catch the off-by-one error, because the substitution is shown rather than assumed.
The sum can be checked without adding anything: the first and last terms average to 25.5, and ten terms at that average give 255. Whenever the partial sum looks wrong, that average-times-count check finds the problem faster than re-adding the list.
A decreasing sequence
Now start at 100 and subtract 7 each step. A negative common difference is entirely ordinary, and the same two formulas apply unchanged.
Inputs First term a₁ = 100, Common difference d = -7, Term number n = 12
- Given a₁ = 100, common difference d = -7, n = 12
- nth term aₙ = a₁ + (n−1)d = 100 + 11·-7 = 23
- Sum of n terms Sₙ = n/2·(a₁ + aₙ) = 12/2·(100 + 23) = 738
Result a₁₂ = 23, Sₙ = 738
The twelfth term is still positive, so the sum keeps growing across this range. Continue far enough and the terms turn negative, after which the partial sum begins to fall — the sum of an arithmetic sequence is not monotonic when d is negative.
The step line shows the multiplication with the negative value in place, which is worth reading: writing 11 × −7 as −77 rather than 77 is where hand calculations on decreasing sequences usually go wrong.
Reading the result
The answer line reports two different things
The subscripted value is the single term at position n. The Sₙ value is the running total of every term from the first to the nth inclusive. Confusing them is easy because they appear together, and for large n they differ by orders of magnitude.
Exactness
With whole-number inputs both outputs are exact integers — no rounding is involved, because the arithmetic is addition and multiplication only. Decimal inputs are displayed to six significant figures, so a common difference such as one third appears rounded.
Why n must be a whole number
The formula would happily evaluate at n = 2.5, but the result would mean nothing: a sequence is defined only at integer positions, and the sum of the first two-and-a-half terms is not a quantity. The solver therefore rejects non-integer n rather than returning a meaningless number.
When you would use this
Totalling a schedule of level increments
A salary rising by a fixed amount each year, or a savings plan increasing the deposit by a set step, is arithmetic. The partial sum gives the cumulative total over any number of years without building a table.
Counting in physical arrangements
Rows of seating that gain a fixed number of seats each row, stacked pipes, or tiles in a stepped pattern are all arithmetic sequences. The nth term gives the size of one row, the sum gives the whole arrangement.
Assumptions and limitations
What this calculator assumes
- The difference between consecutive terms is constant. If it is not, the sequence is not arithmetic and both formulas give wrong answers rather than errors.
- Indexing starts at one, so a₁ is the first term and n counts from there.
- Arithmetic is double precision and displayed to six significant figures; whole-number inputs give exact whole-number results.
Where it stops being the right tool
- Only arithmetic sequences. A sequence multiplying by a constant ratio is geometric and needs the geometric-sequence tool.
- You cannot solve backwards: entering a known term to recover n or d is not supported, only the forward direction.
- There is no infinite sum. An arithmetic series with non-zero d diverges, so only finite partial sums are meaningful.
Common mistakes
Using n instead of n − 1 in the term formula
Why it happens. It feels natural that the tenth term should involve ten of something. In fact it involves nine steps, because the first term is reached with no steps at all.
How to avoid it. Test the formula at n = 1: it must return a₁ exactly. Any version that adds a d at n = 1 is off by one step throughout.
Reading the common difference off the wrong pair
Why it happens. In a sequence written out with a typo, or one that is not actually arithmetic, different pairs give different differences, and whichever pair is checked first becomes d.
How to avoid it. Check at least two consecutive pairs before entering d. If they disagree, the sequence is not arithmetic and this tool is the wrong one for it.
Mistaking the sum for the term
Why it happens. Both numbers appear on the same answer line, and the sum is usually the larger of the two, so it reads like the more important result.
How to avoid it. The subscripted symbol marks the term; Sₙ marks the sum. If you need one seat number, take the term; if you need the whole auditorium, take the sum.
Frequently asked questions
What is the nth term formula?
aₙ = a₁ + (n − 1)d, where a₁ is the first term and d the common difference. The multiplier is n − 1 rather than n because reaching the nth term from the first takes n − 1 steps.
How is the sum calculated?
Sₙ = (n/2)(a₁ + aₙ), the number of terms multiplied by the average of the first and last. It comes from pairing the series with its own reversal, so that every pair adds to the same total.
Can the common difference be negative or zero?
Yes to both. A negative d gives a decreasing sequence and a zero d gives a constant one, where every term equals a₁ and the sum is simply n × a₁. Both are handled by the same formulas.
Why does the solver reject a decimal term number?
Because a sequence is defined only at whole-number positions. There is no term between the second and the third, so a request for term 2.5 has no meaning and is refused rather than answered with a misleading number.