Arithmetic Sequence Calculator
Find the nth term, partial sum, and common difference of any arithmetic sequence. Solve for any unknown and generate up to 1,000 terms.
QUICK PRESETS
What Is the Arithmetic Sequence Calculator?
An arithmetic sequence (also called an arithmetic progression, or AP) is an ordered list of numbers in which the difference between any two consecutive terms is always the same constant, called the common difference. Whether you are counting by twos, calculating loan payments, or modelling linear depreciation, arithmetic sequences show up everywhere.
- ▸Generate mode: Enter a₁, d, and n to instantly see the nth term, partial sum, term list, and a visual bar chart of values.
- ▸Solve for unknown: Know three of {a₁, d, n, aₙ}? Pick the unknown from the tab and fill the rest, the calculator rearranges the formula algebraically.
- ▸Find n for sum: Target a specific partial sum and the calculator solves the resulting quadratic to tell you exactly how many terms you need.
- ▸Formulas panel: Every result shows explicit, sum, and recursive formulas with your numbers substituted in, great for checking homework.
Formula
An arithmetic sequence is fully defined by two numbers: the first term a₁ and the common difference d. From these, every term and any partial sum can be computed directly, no iteration needed.
aₙ = a₁ + (n − 1) × d
Gives the nth term directly. a₁ = first term, d = common difference, n = term number.
Sₙ = n/2 × (a₁ + aₙ)
Sum of first n terms using first and last. Works when you already know aₙ.
Sₙ = n/2 × (2a₁ + (n − 1)d)
Use when you only know a₁ and d, no need to compute aₙ first.
a₁ = given aₙ = aₙ₋₁ + d
Each term is the previous plus d. Useful conceptually but explicit form is faster to compute.
d = (aₙ − a₁) / (n − 1)
Recover d when you know the first term, last term, and the number of terms.
μ = (a₁ + aₙ) / 2 = Sₙ / n
The arithmetic mean equals the average of the first and last term, a property unique to arithmetic sequences.
How to Use
- 1
Choose your mode
Generate Sequence for full output, Solve for Unknown if you are missing one of the four variables, or Find n for Sum if you need to hit a specific total.
- 2
Enter the first term (a₁)
This is the starting value of your sequence. It can be any real number, positive, negative, or fractional.
- 3
Enter the common difference (d)
The fixed amount added to each term. Positive d gives an increasing sequence; negative d gives a decreasing one; d = 0 gives a constant sequence.
- 4
Set the number of terms (n)
In Generate mode, this is how many terms to produce (up to 1,000). In Solve mode it is one of the four known/unknown variables.
- 5
Read your results
The calculator shows aₙ, Sₙ, the arithmetic mean, a full formula breakdown with your values substituted in, and a visual chart for sequences up to 50 terms.
Example Calculation
Example 1 | Standard sequence (a₁ = 3, d = 7, n = 10)
Example 2 | Negative common difference (a₁ = 50, d = −3, n = 15)
A sequence with d < 0 eventually becomes negative if you extend it far enough.
Example 3 | Find common difference (a₁ = 4, a₁₂ = 37, n = 12)
Example 4 | Find n where Sₙ = 820 (a₁ = 1, d = 1)
This is the classic problem of finding how many natural numbers to sum to reach a target.
Understanding Arithmetic Sequence
What Is an Arithmetic Sequence?
An arithmetic sequence, sometimes called an arithmetic progression (AP), is a list of numbers where each term is obtained by adding the same fixed amount to the previous one. That fixed amount is the common difference (d). The sequence 4, 9, 14, 19, 24 has d = 5; the sequence 100, 95, 90, 85 has d = −5. The concept is one of the first topics in algebra precisely because it is the simplest pattern that goes beyond constant values.
Arithmetic vs Geometric vs Harmonic Sequences
| Type | How terms change | Sum convergence | Classic example |
|---|---|---|---|
| Arithmetic | Add constant d | Always diverges | 1, 3, 5, 7, 9 … |
| Geometric | Multiply by ratio r | Converges if |r| < 1 | 1, 2, 4, 8, 16 … |
| Harmonic | Reciprocals of AP | Always diverges | 1, 1/2, 1/3, 1/4 … |
| Fibonacci | Sum of previous two | Diverges | 1, 1, 2, 3, 5, 8 … |
The Four Key Variables
Every arithmetic sequence problem involves four quantities. Know any three and you can find the fourth:
| Variable | Symbol | How to find it |
|---|---|---|
| First term | a₁ | Given, or a₁ = aₙ − (n−1)d |
| Common difference | d | d = (aₙ − a₁) / (n−1) |
| Number of terms | n | n = (aₙ − a₁)/d + 1 |
| nth term | aₙ | aₙ = a₁ + (n−1)d |
| Partial sum | Sₙ | Sₙ = n/2 × (a₁ + aₙ) |
Why the Sum Formula Works
Write the sequence forwards and backwards, then add them term by term:
S = a₁ + (a₁+d) + (a₁+2d) + … + aₙ
S = aₙ + (aₙ−d) + (aₙ−2d) + … + a₁
2S = n × (a₁ + aₙ) → S = n(a₁ + aₙ)/2
Each of the n column pairs sums to (a₁ + aₙ). Writing 2S on the left and dividing by 2 gives the formula. This trick, attributed to Gauss who is said to have solved 1 + 2 + … + 100 = 5050 in seconds as a child, is the most elegant derivation in elementary mathematics.
Finding n for a Target Sum, The Quadratic Approach
Setting Sₙ = target and expanding the formula gives a quadratic in n:
- ▸Sₙ = n/2 × (2a₁ + (n−1)d) = target
- ▸Rearrange: (d/2)n² + (a₁ − d/2)n − target = 0
- ▸Apply the quadratic formula: n = [−(a₁ − d/2) ± √((a₁−d/2)² + 2d×target)] / d
- ▸Take the positive root and round up to the nearest integer.
The calculator handles all of this automatically in the "Find n for Sum" tab.
Real-World Applications of Arithmetic Sequences
- ▸Straight-line depreciation: A machine worth $50,000 depreciating $4,000/year forms an AP with d = −4000. After n years, its value is aₙ₊₁ = 50000 − 4000n.
- ▸Equal instalment savings: Depositing a fixed amount each month creates an AP. The total saved after n months is Sₙ = n × monthly deposit (if the rate is 0).
- ▸Stadium seating: If row 1 has 20 seats and each row adds 3 more, row k has 20 + (k−1)×3 seats. Total capacity over n rows = Sₙ.
- ▸Physics, uniform acceleration: Distances covered in successive equal time intervals under constant acceleration form an AP (d = acceleration × time²).
- ▸Music, equal temperament: The 12 semitones in an octave are spaced by equal frequency ratios, but their names (C, C#, D …) form an AP of indices.
- ▸Programming, loop counters: for (i = 0; i < n; i += step) generates an arithmetic sequence of index values.
Arithmetic Progressions in Number Theory
Arithmetic progressions play a central role in advanced mathematics. Dirichlet's theorem (1837) states that any AP with gcd(a₁, d) = 1 contains infinitely many prime numbers. The celebrated Green-Tao theorem (2004) proved that the prime numbers themselves contain arbitrarily long arithmetic progressions, a result that earned a Fields Medal. Even at the school level, the fact that the sum of consecutive odd numbers is always a perfect square (1, 1+3=4, 1+3+5=9, …) is an arithmetic sequence identity with a beautiful visual proof.
Common Mistakes to Avoid
- ▸Using (n) instead of (n−1) in the nth term formula: aₙ = a₁ + (n−1)d, not a₁ + nd.
- ▸Forgetting that d can be zero (constant sequence) or negative (decreasing sequence).
- ▸Confusing aₙ (nth term value) with n (the term's position index).
- ▸Applying the partial sum formula to a geometric sequence, the formulas are different.
- ▸Assuming every integer solution to the quadratic (for the sum problem) is valid, check that n ≥ 1.
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence is an ordered list of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference (d). Examples: 2, 5, 8, 11 (d = 3); 10, 7, 4, 1 (d = −3); 5, 5, 5 (d = 0).
What is the formula for the nth term of an arithmetic sequence?
The nth term is given by aₙ = a₁ + (n − 1) × d, where a₁ is the first term, d is the common difference, and n is the term's position. For example, in 3, 7, 11, 15 … (a₁ = 3, d = 4), the 10th term is 3 + 9 × 4 = 39.
How do you find the sum of an arithmetic series?
Use the Gauss formula: Sₙ = n/2 × (a₁ + aₙ), or equivalently Sₙ = n/2 × (2a₁ + (n−1)d). Both give the same result. The insight is that pairing the first and last term, second and second-to-last, etc. always gives the same sum, and there are n/2 such pairs.
How is an arithmetic sequence different from a geometric sequence?
In an arithmetic sequence, terms change by a constant addition (d). In a geometric sequence, terms change by a constant multiplication (common ratio r). The sum formula differs: an infinite arithmetic series always diverges, while a geometric series converges when |r| < 1.
What does a negative common difference mean?
It means the sequence is decreasing, each term is smaller than the one before it. For example, a₁ = 20, d = −4 gives 20, 16, 12, 8, 4, 0, −4, …. The sequence will eventually pass through zero and continue into negative values.
How do I find n if I know the nth term but not the position?
Rearrange the explicit formula: n = (aₙ − a₁)/d + 1. For example, if a₁ = 3, d = 5, and aₙ = 48, then n = (48 − 3)/5 + 1 = 10. The calculator's "Solve for Unknown" mode does this automatically. Note: the result must be a positive integer, otherwise the value doesn't belong to the sequence.
Can the common difference be a fraction or decimal?
Yes. d can be any real number, integer, fraction, or decimal. For example, a₁ = 0, d = 0.5 gives 0, 0.5, 1, 1.5, 2, … This is a perfectly valid arithmetic sequence used in contexts like half-step musical scales or time series with equal intervals.
What is the arithmetic mean of a sequence?
For an arithmetic sequence, the mean (average) of all n terms always equals the average of the first and last term: μ = (a₁ + aₙ)/2. This is also equal to Sₙ/n. The median and mean of an arithmetic sequence are always the same.
Where are arithmetic sequences used in real life?
They appear in: straight-line depreciation (an asset losing $500/year), equal instalment savings plans, equally-spaced time intervals in physics, taxi fare structures (fixed base + per-km charge), musical intervals (semitones), and seating arrangements in auditoriums where each row has one more seat than the last.
What is the Green-Tao theorem about arithmetic progressions?
The Green-Tao theorem (2004) proved that the prime numbers contain arithmetic progressions of any finite length, meaning you can always find k primes in a row with a common difference d, for any k. This is a deep result connecting arithmetic sequences to number theory.