Math

Arithmetic Sequence Calculator — nth Term & Sum

Q: What is the formula for the nth term of an arithmetic sequence?

The nth term is **aₙ = a₁ + (n − 1) · d**, where a₁ is the first term, d is the common difference, and n is the 1-indexed position. Example: a₁ = 2, d = 3 → a₁₀ = 2 + 9 · 3 = **29**. The (n − 1) factor is critical — the difference is applied one fewer time than the term number because the first term itself requires zero additions. A common exam mistake is writing aₙ = a₁ + n · d, which gives a result one step too large.

Q: How do I calculate the sum of an arithmetic sequence?

Use **Sₙ = n/2 · (a₁ + aₙ)** or equivalently **Sₙ = n/2 · (2a₁ + (n − 1) · d)**. Both are identical — substitute aₙ = a₁ + (n − 1) · d into the first to get the second. For the classic Gauss problem (1 to 100): S₁₀₀ = 100/2 · (1 + 100) = 50 · 101 = **5,050**. The formula works because every pair of terms equidistant from the ends sums to the same constant.

Q: Can the common difference d be zero or negative?

Yes. If **d = 0**, every term equals a₁ and Sₙ = n · a₁ (a constant sequence). If **d < 0**, the sequence is strictly decreasing — for example a₁ = 10, d = −2 gives 10, 8, 6, 4, 2, 0, −2, … The same formulas apply without modification; just substitute the signed value of d. Always preserve the sign — replacing d with |d| turns a decreasing sequence into an increasing one.

Q: What is the difference between an arithmetic and a geometric sequence?

An **arithmetic sequence** adds a fixed constant each step (e.g., 2, 5, 8, 11 — adds 3). A **geometric sequence** multiplies by a fixed ratio each step (e.g., 2, 6, 18, 54 — multiplies by 3). Test: compute consecutive differences — if they're equal, it's arithmetic; compute consecutive ratios — if they're equal, it's geometric. The sum formulas are completely different: Sₙ(AP) = n/2·(a₁+aₙ) vs. Sₙ(GP) = a₁·(rⁿ − 1)/(r − 1).

Q: How do I find which term position n equals a given value?

Rearrange the nth term formula: **n = (aₙ − a₁) / d + 1**. Example: which term equals 50 in 2, 5, 8, …? n = (50 − 2)/3 + 1 = 16 + 1 = **17**. Check: a₁₇ = 2 + 16·3 = 50 ✓. If the result is not a positive integer, that value is not in the sequence.

Q: What is the arithmetic mean of the terms in an arithmetic sequence?

The average of all n terms equals the midpoint of the first and last: **mean = (a₁ + aₙ) / 2**. For 2, 5, 8, …, 29 (n = 10): mean = (2 + 29)/2 = **15.5**, consistent with S₁₀/10 = 155/10 = 15.5. When n is odd, the mean equals the literal middle term. When n is even, it falls between the two middle terms and may not appear in the sequence.

Q: How does an arithmetic sequence relate to linear functions?

An AP is a **linear function evaluated at positive integers**: f(n) = a₁ + (n − 1)·d is linear in n with slope d. Plotting the terms (n, aₙ) gives points on a straight line. The sum Sₙ corresponds to the area of a trapezoid under that line — which is why Sₙ = n/2·(a₁ + aₙ) has the same structure as the trapezoid area formula: (width) × (average height).

Q: Do arithmetic sequences appear on the SAT or GRE?

Yes. Both exams regularly include AP problems, often disguised as word problems. Typical SAT question: 'The first term is 4 and each term is 6 more than the previous. What is the sum of the first 8 terms?' Solution: a₈ = 4 + 7·6 = 46; S₈ = 8/2·(4+46) = 4·50 = **200**. Mastering the two formulas (aₙ and Sₙ) covers the vast majority of such questions.

Q: What if I know Sₙ and need to find n?

Rearranging Sₙ = n/2·(2a₁ + (n−1)·d) gives a quadratic in n: **d·n² + (2a₁ − d)·n − 2Sₙ = 0**. Solve with the quadratic formula and take the positive integer root. Example: a₁ = 5, d = 2, Sₙ = 60 → 2n² + 8n − 120 = 0 → n = **6**. Verify: terms are 5, 7, 9, 11, 13, 15; sum = 60 ✓.

Q: What is the sum of the first n odd numbers?

The odd numbers form an AP with a₁ = 1 and d = 2, so Sₙ = n/2·(2·1 + (n−1)·2) = n/2·(2n) = **n²**. The first n odd numbers always sum to a perfect square: 1+3+5+7 = 4² = 16; 1+3+5+7+9 = 5² = 25. This is a classic result used in competition math and proves geometrically that n² equals the count of unit squares in an n×n grid.

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Reviewed by: Martín Rodríguez (política editorial ) · Last reviewed: 4 jun 2026

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An arithmetic sequence (also called an arithmetic progression, or AP) is a list of numbers where each term increases or decreases by a fixed amount called the common difference (d). Classic example: 2, 5, 8, 11, 14 — where d = 3. Arithmetic sequences are a core topic in algebra and appear on every standardized test from the SAT and ACT to the GRE and GMAT. This calculator finds the two key values for any AP: the n-th term (aₙ = a₁ + (n − 1) · d) and the sum of the first n terms (Sₙ = n/2 · (a₁ + aₙ)). Enter any three inputs — first term a₁, common difference d, and position n — and get both results instantly with a step-by-step breakdown. The sum formula traces back to a famous anecdote about Carl Friedrich Gauss, who summed 1 through 100 as a schoolchild by pairing 1+100, 2+99, … each summing to 101 across 50 pairs — total 5,050. The same pairing trick works for any AP: Sₙ = n × (average of first and last term). Arithmetic sequences model countless real-world situations: salary schedules with annual raises, loan principal payments, theater seating where each row adds a fixed number of seats, and constant-acceleration kinematics where distance covered each second forms an AP.

Last reviewed: June 3, 2026 Verified by Martín Rodríguez Source: Wikipedia – Arithmetic Progression, Khan Academy – Arithmetic Sequences, Math Is Fun – Arithmetic Sequences and Sums, Brilliant – Arithmetic Progressions 100% private

For an arithmetic sequence with first term a₁, common difference d, and n terms: **aₙ = a₁ + (n − 1) · d** and **Sₙ = n/2 · (a₁ + aₙ)**. Example: a₁ = 2, d = 3, n = 10 → a₁₀ = 29, S₁₀ = 155.

When to use this calculator

Finding the 10th term of the sequence 2, 5, 8, 11, … (a₁ = 2, d = 3) to verify a pattern on an SAT math exam — answer: a₁₀ = 29, S₁₀ = 155.
Calculating the total distance traveled by an object that moves 5 m in the 1st second and gains 2 m each subsequent second over 10 seconds — a direct AP sum: S₁₀ = 140 m.
Summing 12 monthly salary increments where a base of $3,000 grows by $150 each month to budget total annual payroll cost: S₁₂ = $45,900 (vs. $36,000 if flat).
Determining the seating capacity of a theater where the first row has 20 seats and each subsequent row adds 4 seats across 30 rows: a₃₀ = 136 seats; total = 2,340 seats.
Computing the total number of bricks needed to build a triangular wall where the bottom row has 50 bricks and each row above has 1 less, until the top row has 1 brick: n = 50, S₅₀ = 1,275 bricks.
Modeling a loan with constant principal payment of $500/month against a starting balance of $24,000 at zero interest: balance after n months follows an AP with a₁ = $23,500 and d = −$500; the loan zeros out at n = 48.
Finding the 25th odd number for a programming exercise: the odd-number AP has a₁ = 1, d = 2, so a₂₅ = 1 + 24 · 2 = 49, and the sum of the first 25 odd numbers = 25² = 625.
Calculating the cumulative depreciation of an asset using straight-line schedule of $1,200/year over 7 years (d = 0, constant): S₇ = $8,400 total accumulated depreciation.

Worked Example

Inputs: a₁ = 2, d = 3, n = 10
n-th term: a₁₀ = 2 + (10 − 1) · 3 = 2 + 27 = 29
Sum: S₁₀ = 10/2 · (2 + 29) = 5 · 31 = 155

Result: a₁₀ = 29, S₁₀ = 155

How it works

2 min read

Formulas Used

An arithmetic sequence is fully defined by three values: first term (a₁), common difference (d), and number of terms (n).

// n-th term
aₙ = a₁ + (n − 1) · d

// Sum of the first n terms (two equivalent forms)
Sₙ = n/2 · (a₁ + aₙ)          // using first and last term
Sₙ = n/2 · (2·a₁ + (n−1)·d)   // using a₁ and d only

Both sum formulas are identical — substitute aₙ = a₁ + (n−1)·d into the first to derive the second.

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Reference Table: Common Arithmetic Sequences

Use this table to verify your results at a glance:

a₁	d	n	aₙ	Sₙ	Sequence preview
1	1	10	10	55	1, 2, 3, …, 10
2	3	10	29	155	2, 5, 8, …, 29
0	5	8	35	140	0, 5, 10, …, 35
10	−2	6	0	30	10, 8, 6, 4, 2, 0
3	7	5	31	85	3, 10, 17, 24, 31
100	25	12	375	2,850	100, 125, 150, …, 375
1	2	50	99	2,500	1, 3, 5, …, 99 (odd numbers)
1	1	100	100	5,050	1, 2, 3, …, 100 (Gauss)

> Classic result: Sum of the first 100 integers = 5,050 (Gauss). Sum of first n positive integers = n(n+1)/2.

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Step-by-Step Examples

Example 1 — Increasing sequence (positive d)

Inputs: a₁ = 2, d = 3, n = 10

Step 1 – n-th term: a₁₀ = 2 + (10 − 1) · 3 = 2 + 27 = 29

Step 2 – Sum: S₁₀ = 10/2 · (2 + 29) = 5 · 31 = 155

Example 2 — Decreasing sequence (negative d)

Inputs: a₁ = 50, d = −4, n = 8

Step 1 – n-th term: a₈ = 50 + (8 − 1) · (−4) = 50 − 28 = 22

Step 2 – Sum: S₈ = 8/2 · (50 + 22) = 4 · 72 = 288

Example 3 — Theater seating (real-world)

Inputs: a₁ = 20 seats, d = 4, n = 30 rows

Step 1 – n-th term: a₃₀ = 20 + 29 · 4 = 136 seats in last row

Step 2 – Sum: S₃₀ = 30/2 · (20 + 136) = 15 · 156 = 2,340 total seats

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Common Mistakes

1. Off-by-one — Writing aₙ = a₁ + n·d instead of a₁ + (n−1)·d shifts every result one position forward.
2. Confusing aₙ with Sₙ — The calculator returns both; always check which value the problem asks for.
3. Sign errors with negative d — Always substitute d = −|d|; don't drop the negative sign.
4. Counting n from 0 — n is 1-indexed: the first term is at n = 1, not n = 0.
5. Mixing AP and GP — Arithmetic adds a constant; geometric multiplies by a constant. The formulas are completely different.

Frequently asked questions

What is the formula for the nth term of an arithmetic sequence?

The nth term is aₙ = a₁ + (n − 1) · d, where a₁ is the first term, d is the common difference, and n is the 1-indexed position. Example: a₁ = 2, d = 3 → a₁₀ = 2 + 9 · 3 = 29. The (n − 1) factor is critical — the difference is applied one fewer time than the term number because the first term itself requires zero additions. A common exam mistake is writing aₙ = a₁ + n · d, which gives a result one step too large.

How do I calculate the sum of an arithmetic sequence?

Use Sₙ = n/2 · (a₁ + aₙ) or equivalently Sₙ = n/2 · (2a₁ + (n − 1) · d). Both are identical — substitute aₙ = a₁ + (n − 1) · d into the first to get the second. For the classic Gauss problem (1 to 100): S₁₀₀ = 100/2 · (1 + 100) = 50 · 101 = 5,050. The formula works because every pair of terms equidistant from the ends sums to the same constant.

Can the common difference d be zero or negative?

Yes. If d = 0, every term equals a₁ and Sₙ = n · a₁ (a constant sequence). If d < 0, the sequence is strictly decreasing — for example a₁ = 10, d = −2 gives 10, 8, 6, 4, 2, 0, −2, … The same formulas apply without modification; just substitute the signed value of d. Always preserve the sign — replacing d with |d| turns a decreasing sequence into an increasing one.

What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds a fixed constant each step (e.g., 2, 5, 8, 11 — adds 3). A geometric sequence multiplies by a fixed ratio each step (e.g., 2, 6, 18, 54 — multiplies by 3). Test: compute consecutive differences — if they're equal, it's arithmetic; compute consecutive ratios — if they're equal, it's geometric. The sum formulas are completely different: Sₙ(AP) = n/2·(a₁+aₙ) vs. Sₙ(GP) = a₁·(rⁿ − 1)/(r − 1).

How do I find which term position n equals a given value?

Rearrange the nth term formula: n = (aₙ − a₁) / d + 1. Example: which term equals 50 in 2, 5, 8, …? n = (50 − 2)/3 + 1 = 16 + 1 = 17. Check: a₁₇ = 2 + 16·3 = 50 ✓. If the result is not a positive integer, that value is not in the sequence.

How do I find the common difference d if I only know two terms?

Given terms at positions m and n: d = (aₙ − aₘ) / (n − m). Example: if the 3rd term is 11 and the 7th term is 27 → d = (27 − 11)/(7 − 3) = 16/4 = 4. Then recover a₁ = aₘ − (m − 1)·d = 11 − 2·4 = 3, giving the sequence 3, 7, 11, 15, 19, 23, 27, …

What is the arithmetic mean of the terms in an arithmetic sequence?

The average of all n terms equals the midpoint of the first and last: mean = (a₁ + aₙ) / 2. For 2, 5, 8, …, 29 (n = 10): mean = (2 + 29)/2 = 15.5, consistent with S₁₀/10 = 155/10 = 15.5. When n is odd, the mean equals the literal middle term. When n is even, it falls between the two middle terms and may not appear in the sequence.

How does an arithmetic sequence relate to linear functions?

An AP is a linear function evaluated at positive integers: f(n) = a₁ + (n − 1)·d is linear in n with slope d. Plotting the terms (n, aₙ) gives points on a straight line. The sum Sₙ corresponds to the area of a trapezoid under that line — which is why Sₙ = n/2·(a₁ + aₙ) has the same structure as the trapezoid area formula: (width) × (average height).

Do arithmetic sequences appear on the SAT or GRE?

Yes. Both exams regularly include AP problems, often disguised as word problems. Typical SAT question: 'The first term is 4 and each term is 6 more than the previous. What is the sum of the first 8 terms?' Solution: a₈ = 4 + 7·6 = 46; S₈ = 8/2·(4+46) = 4·50 = 200. Mastering the two formulas (aₙ and Sₙ) covers the vast majority of such questions.

What if I know Sₙ and need to find n?

Rearranging Sₙ = n/2·(2a₁ + (n−1)·d) gives a quadratic in n: d·n² + (2a₁ − d)·n − 2Sₙ = 0. Solve with the quadratic formula and take the positive integer root. Example: a₁ = 5, d = 2, Sₙ = 60 → 2n² + 8n − 120 = 0 → n = 6. Verify: terms are 5, 7, 9, 11, 13, 15; sum = 60 ✓.

What is the sum of the first n odd numbers?

The odd numbers form an AP with a₁ = 1 and d = 2, so Sₙ = n/2·(2·1 + (n−1)·2) = n/2·(2n) = n². The first n odd numbers always sum to a perfect square: 1+3+5+7 = 4² = 16; 1+3+5+7+9 = 5² = 25. This is a classic result used in competition math and proves geometrically that n² equals the count of unit squares in an n×n grid.