Arithmetic Sequence Calculator

An arithmetic sequence adds the same fixed amount to every term. The nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n/2 · (a₁ + aₙ). Example: starting at 5 with a common difference of 3, the 20th term is 62 and the first 20 terms add up to 670.

Before you start, check that your sequence really is arithmetic. Subtract each term from the one that follows it. If every gap is the same number, this page is the right tool. If every ratio is the same instead, the sequence is geometric and grows by compounding, which the Compound Interest Calculator models. If each term is the sum of the two before it, you have the Fibonacci pattern and want the Fibonacci Calculator instead.

What an arithmetic sequence actually is

A sequence is just an ordered list of numbers, and each entry is called a term. The first term is written a₁, the second a₂, and the term in position n is written aₙ. A sequence is arithmetic when the difference between any two neighboring terms is constant. That constant is the common difference d, and you find it by subtracting any term from the one immediately after it.

Take 5, 8, 11, 14, 17, 20. Subtracting gives 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3, and so on. Every gap is 3, so the sequence is arithmetic with d = 3. Plotted on a graph, the terms sit on a perfectly straight line, which is why arithmetic sequences are also described as showing linear growth. The common difference is the slope of that line.

The name arithmetic progression (often abbreviated AP) means exactly the same thing and is the more common wording in British and Indian textbooks. When you add the terms of an arithmetic sequence together you get an arithmetic series, which is what the sum formula computes.

The two formulas you need

Everything on this page comes from two exact identities. The first gives any single term:

Read it as: start at the first term, then take n − 1 steps of size d. The reason it is n − 1 rather than n is that the first term requires no steps at all. Term 2 is one step away from term 1, term 3 is two steps away, and term n is n − 1 steps away.

The second identity adds up the first n terms:

In words: the average of the first and last term, multiplied by how many terms there are. If you do not yet know the last term, substitute the first formula into the second to get the equivalent one-step version Sₙ = n/2 · (2a₁ + (n − 1)d). Both give identical answers; the calculator uses the first and shows the arithmetic in its step list.

A third rearrangement is useful when you know two terms but not the difference:

How to use this calculator

1. Pick your mode. Choose I know a₁ and d when the problem states the first term and the common difference. Choose I know two terms when you are told, for example, that the 4th term is 14 and the 9th term is 29.
2. Enter the numbers. Negatives and decimals are fine for the first term and the difference. Term positions must be whole numbers of 1 or more.
3. Set n. This is the term you want. It also controls how many terms are summed, so n = 20 gives both the 20th term and the total of terms 1 through 20.
4. Read the result card. The headline number is aₙ. The tiles below carry the sum, the first term, the common difference and the average of the terms.
5. Study the rule and the table. The explicit and recursive formulas are written out for your exact numbers, and the table lists every term with a running total so you can verify the sum by hand.

Worked example 1: the 20th term and the sum

A sequence starts at 5 and increases by 3 each step: 5, 8, 11, 14, 17, and so on. Find the 20th term and the sum of the first 20 terms.

• Known values: a₁ = 5, d = 3, n = 20.
• Steps taken: n − 1 = 19, so the total climb is 19 × 3 = 57.
• 20th term: a₂₀ = 5 + 57 = 62.
• Sum: S₂₀ = 20/2 × (5 + 62) = 10 × 67 = 670.
• Check: the average term is 670 ÷ 20 = 33.5, and (5 + 62) ÷ 2 = 33.5. The two agree, so the sum is right.

The explicit rule for this sequence simplifies to aₙ = 3n + 2, because 5 + (n − 1) × 3 = 3n + 2. That simplified form is what the calculator prints beside the raw formula, and it makes the linear nature obvious: slope 3, intercept 2.

Term-by-term table: 5, 8, 11, 14, and so on

Here are the first twelve terms of the same sequence with a running sum, so you can see both formulas working line by line:

Notice that the running sum grows faster and faster even though the terms themselves grow by a steady 3. That is because each new term added is larger than the last, so an arithmetic series is quadratic even when the sequence is linear. Continuing the same sequence, the 50th term is 152 and the first 50 terms total 3,925; the 100th term is 302 and the first 100 terms total 15,350.

Worked example 2: finding the common difference from two terms

A problem tells you only that the 4th term is 14 and the 9th term is 29. Find the rule, the 20th term and the sum of the first 20 terms.

• Gap in position: 9 − 4 = 5 steps. Gap in value: 29 − 14 = 15.
• Common difference: d = 15 ÷ 5 = 3.
• Back up to the first term: a₁ = 14 − (4 − 1) × 3 = 14 − 9 = 5.
• 20th term: a₂₀ = 5 + 19 × 3 = 62.
• Sum: S₂₀ = 20/2 × (5 + 62) = 670.

This is the same sequence as example 1, described differently. That is the point of the two-term mode: whichever two terms a problem hands you, the sequence is fully determined, because two points define a line.

Common difference from two terms: six solved cases

Each row applies d = (aₖ − aₘ) / (k − m) and then a₁ = aₘ − (m − 1)d, followed by the 20th term and the 20-term sum:

The fifth row is worth a second look. When two different positions hold the same value, the common difference is 0 and the sequence is constant, so the sum is simply 20 × 18 = 360. The fourth row shows a negative difference derived from a falling pair of terms.

Worked example 3: a decreasing sequence

Not every arithmetic sequence grows. Suppose a machine starts with 100 units of coolant and loses 7 units per cycle: 100, 93, 86, 79, and so on. What is the level after 15 cycles, and what is the running total of all fifteen readings?

• Known values: a₁ = 100, d = −7, n = 15.
• 15th term: a₁₅ = 100 + 14 × (−7) = 100 − 98 = 2.
• Sum of the fifteen readings: S₁₅ = 15/2 × (100 + 2) = 7.5 × 102 = 765.
• The 10th term is 37 and the first ten readings total 685, so cycles 11 through 15 add another 80.

Because d is negative, the terms eventually pass zero and turn negative: the 20th term is −33. Solving 0 = 100 + (n − 1)(−7) gives n = 15.29, which is not a whole number, so no term is exactly zero. The sequence steps straight from 2 at term 15 to −5 at term 16.

Eight sequences with their rule, 10th term and 10-term sum

Use this as an answer key for practice. Every row was produced with the same two formulas:

Two rows are famous special cases. The counting numbers 1, 2, 3 give the triangular numbers as their sums, so Sₙ = n(n + 1)/2 and adding 1 through 100 gives 5,050. The odd numbers 1, 3, 5, 7 have sums that are perfect squares: the first ten odd numbers total 100, which is 10².

Why the sum formula works: the pairing trick

Write the series forward and then backward under itself. Adding the two lines column by column pairs the first term with the last, the second with the second-to-last, and so on. Every one of those pairs adds to exactly the same total, a₁ + aₙ, because whatever one term gains the other loses. There are n columns, so twice the sum equals n × (a₁ + aₙ), and halving gives Sₙ = n/2 · (a₁ + aₙ).

The classic demonstration is adding 1 through 100. Pair 1 with 100, 2 with 99, 3 with 98, and so on: fifty pairs, each adding to 101, so the total is 50 × 101 = 5,050. The same reasoning gives 1 through 50 as 1,275. This is the fastest mental check on any sum this calculator produces: the average of the terms must equal the midpoint of the first and last term.

Which term is a given value? Solving for n

Rearranging the nth term formula gives n = (aₙ − a₁) / d + 1. Applied to the sequence 5, 8, 11, 14 with d = 3:

A non-integer answer is not an error. It simply means the value falls between two terms and is never reached exactly. Note also that n must be positive: a value below the first term cannot appear in an increasing sequence.

Arithmetic versus geometric: adding against multiplying

The single most useful contrast is between adding a constant and multiplying by a constant. Both sequences below start at 100. The arithmetic one adds 10 per step; the geometric one multiplies by 1.10 per step:

The two start together and stay close for a few steps, then the geometric column pulls away and never comes back. That gap is the whole difference between simple interest, which is arithmetic, and compound interest, which is geometric. If your problem involves a fixed percentage per period rather than a fixed amount, this page is the wrong tool.

Where arithmetic sequences show up in real life

• Escalating savings plans. Deposit $50 in week 1 and add $10 more each week. Week 52 deposits $560, and the year total is $15,860, since S₅₂ = 52/2 × (50 + 560).
• Seating and stadium rows. Row 1 has 20 seats and each row adds 2. Row 15 has 48 seats and the whole 15-row section holds 510 people.
• Training plans. Run 2 miles in week 1 and add half a mile weekly. Week 12 is a 7.5 mile run, and the twelve weeks total 57 miles.
• Straight-line depreciation. An asset that loses the same dollar amount every year forms a decreasing arithmetic sequence.
• Simple interest. A balance that earns the same dollar interest each year grows arithmetically, unlike a compounding balance.
• Stacked objects. Logs, pipes or cans stacked with one fewer per layer form the classic arithmetic pile problem.

Key terms explained

• Term: one entry in the list. The nth term is written aₙ.
• First term (a₁): the starting value; everything else is measured from it.
• Common difference (d): the fixed amount added at each step. It can be positive, negative, fractional or zero.
• Explicit formula: a rule that gives any term directly from its position, aₙ = a₁ + (n − 1)d.
• Recursive formula: a rule that gives each term from the previous one, aₙ = aₙ₋₁ + d.
• Arithmetic series: the sum of the terms, written Sₙ. Also called a partial sum.
• Arithmetic progression (AP): another name for an arithmetic sequence.
• Arithmetic mean: the average of two values. In an arithmetic sequence every term is the arithmetic mean of its two neighbors.

What changes the result

Four inputs drive everything on this page, and each one moves the answer in a predictable direction:

• The first term shifts every term and the whole sum. Raising a₁ by 1 raises the sum of n terms by exactly n.
• The common difference sets the slope. Raising d by 1 raises the nth term by n − 1 and the sum by n(n − 1)/2, which is why a small change in d has an outsized effect on a long sum.
• The number of terms drives the sum quadratically. Doubling n roughly quadruples the sum when d dominates the first term.
• The sign of d decides whether the sequence climbs, falls or stays flat, and whether the sum can turn negative partway along.

Tips for getting it right

• Confirm the sequence really is arithmetic by checking at least three consecutive gaps, not just one.
• Count steps, not terms: reaching term n takes n − 1 steps of size d.
• Keep the sign of d attached to it. A decreasing sequence has a negative common difference all the way through the arithmetic.
• Verify a sum by dividing it by n. The result must equal the midpoint of the first and last term.
• When a problem gives two arbitrary terms, use the position gap as the divisor, not the smaller position.
• For a sum that starts partway along, compute S for the whole range and subtract S for the part you are skipping.

Limitations of this calculator

This page covers finite arithmetic sequences only. It does not handle geometric or harmonic sequences, quadratic patterns, Fibonacci-style recursions, infinite series, or sums of squares and cubes. Results are shown as decimals rounded to six places rather than exact fractions, so a repeating common difference such as one third displays as 0.333333. There is no graphing, and the term table is capped at 25 rows plus the highlighted target term to keep long sequences readable. For sums that begin at a term other than the first, run the calculator twice and subtract.

How this compares to related calculators

This page is for sequences that grow by a fixed amount. Reach for a sibling tool when the pattern is different: the Fibonacci Calculator when each term is the sum of the two before it, the Compound Interest Calculator when growth is a fixed percentage per period rather than a fixed amount, the Exponent Calculator when you need the powers behind that geometric growth, the Average Calculator when you want the mean, median and mode of an arbitrary list instead of a patterned one, and the Scientific Calculator for one-off arithmetic on a term you already have.

• Checking whether a list of test scores has an average, not a pattern? Use the Average Calculator.
• Modeling money that compounds instead of stepping up by a flat amount? Use the Compound Interest Calculator.
• Need the spread of a data set rather than its rule? The Standard Deviation Calculator handles variance and standard deviation.
• Working with factorials or permutations from the same chapter? The Factorial Calculator covers n! directly.