Arithmetic Progression Calculator
Enter the number of terms, the first term, and the common difference to generate the full arithmetic progression and the sum of its first n terms instantly.
Solve Arithmetic Progressions
Arithmetic Progression Formula
Where:
- \(a_1\) — the first term of the sequence
- \(d\) — the common difference between consecutive terms
- \(n\) — the number of terms
- \(a_n\) — the n-th term
- \(S_n\) — the sum of the first \(n\) terms
The first formula tells you the value of any term without listing the whole sequence; the second adds all terms at once. The sum formula also equals \(S_n = \frac{n}{2}(a_1 + a_n)\), i.e. the number of terms times the average of the first and last term.
Arithmetic Progression – Calculation Example
Example: find the first 6 terms of an arithmetic progression that starts at 5 with a common difference of 2, and their sum.
\[ a_n = a_1 + (n - 1)d = 5 + (6 - 1) \cdot 2 = 15 \]
\[ S_6 = \frac{6}{2}\left[2 \cdot 5 + (6 - 1) \cdot 2\right] = 3 \cdot 20 = 60 \]
Result: the progression is 5, 7, 9, 11, 13, 15 and the sum of the first 6 terms is 60.
How to Use the Calculator
- Enter the nᵗʰ term field — this is how many terms of the progression you want to generate (e.g. 6).
- Enter the First term \(a_1\), the value the sequence starts from (e.g. 5).
- Enter the Common difference \(d\), the constant added to each term (e.g. 2). Use a negative value for a decreasing sequence.
- Read the results: Arithmetic progression lists all \(n\) terms, and Sum of first n terms gives their total.
What Is an Arithmetic Progression?
An arithmetic progression (AP), also called an arithmetic sequence, is a list of numbers in which each term differs from the previous one by the same constant \(d\), the common difference. The sequence 2, 5, 8, 11, 14, … is arithmetic because every step adds 3; the sequence 2, 4, 8, 16, … is not, because the ratio (not the difference) is constant — that one is a geometric pattern.
Two values fully define an AP: the first term \(a_1\) and the common difference \(d\). From them you can reconstruct any term with \(a_n = a_1 + (n-1)d\), which means you can find the 100th term without writing out the 99 before it. The sign of \(d\) decides the behavior: a positive difference makes the sequence grow, a negative one makes it shrink, and \(d = 0\) makes every term identical.
The sum \(S_n\) answers questions like “how much in total after n deposits or payments?” Thanks to the identity \(S_n = \frac{n}{2}(a_1 + a_n)\), the sum is simply the average of the first and last term multiplied by the count — the same idea behind an average calculation. For example, with terms 5 through 15 the average of the endpoints is 10, and \(6 \times 10 = 60\).
Arithmetic progressions appear in many practical settings: equal monthly savings or installments, straight-line depreciation, seating rows that grow by a fixed number of seats, and ladder-style discounts or pay raises. Whenever a quantity changes by the same absolute amount each step, an AP describes it.
Tips, Assumptions and Limitations
- Inputs can be decimals or negative numbers. The default step is 1, but you can type values like −3 or 0.5; a fractional difference still produces a valid AP.
- Large n produces long output. The calculator lists every term up to n, so very large values of n will return a very long list; the sum stays exact regardless.
- Indexing starts at 1. Here \(a_1\) is the first term and \(a_n\) is the term in position n, the standard convention in school mathematics.
- Verify a sequence you already have: subtract consecutive terms; if the differences are all equal to \(d\), the sequence is arithmetic. If instead the ratios are constant, you need geometric tools rather than this calculator.
- For related sums over number patterns, see the sum of consecutive squares or Floyd's triangle calculators.
Frequently Asked Questions
How do I find the common difference of an arithmetic sequence?
Subtract any term from the term that follows it: \(d = a_{n+1} - a_n\). For example, in 7, 12, 17, 22 you get \(12 - 7 = 5\), so \(d = 5\). Check the subtraction on at least one more pair of consecutive terms — they must all give the same value, otherwise the sequence is not arithmetic.
How do I calculate the sum without knowing the last term?
Use \(S_n = \frac{n}{2}\left[2a_1 + (n-1)d\right]\), which needs only the first term, the difference, and the count. Alternatively, compute the last term first with \(a_n = a_1 + (n-1)d\), then apply \(S_n = \frac{n}{2}(a_1 + a_n)\). Both routes give exactly the same result, so pick whichever needs fewer steps for your data.
How do I find how many terms there are if I know the first and last term?
Rearrange the n-th term formula: \(n = \frac{a_n - a_1}{d} + 1\). For instance, an AP from 3 to 39 with \(d = 4\) has \(\frac{39 - 3}{4} + 1 = 10\) terms. Don't forget the +1 — forgetting it is the single most common mistake here, because the count includes both endpoints.
Can the common difference be negative or a fraction?
Yes. A negative \(d\) simply produces a decreasing sequence, e.g. \(20, 15, 10, 5\) with \(d = -5\). A fractional difference such as \(d = 0.5\) is equally valid: 2, 2.5, 3, 3.5, … Only \(d = 0\) is a special case — every term equals the first term and the sequence never changes.
What is the sum of the first n natural numbers?
It is the classic AP sum with \(a_1 = 1\) and \(d = 1\): \(S_n = \frac{n(n+1)}{2}\). So \(1 + 2 + \dots + 100 = \frac{100 \cdot 101}{2} = 5050\), the result famously attributed to young Gauss. This shortcut works for any sequence of consecutive integers, not just starting from 1.
How is an arithmetic progression different from a geometric one?
In an arithmetic progression each term is the previous term plus a constant, so consecutive differences are equal. In a geometric progression each term is the previous term times a constant, so consecutive ratios are equal. Quick test: 3, 6, 9, 12 is arithmetic (difference 3), while 3, 6, 12, 24 is geometric (ratio 2). Use the geometric tools for the second pattern.
Can I use this calculator for real-world problems like savings or depreciation?
Yes, whenever a quantity changes by the same absolute amount each period. Equal monthly deposits, straight-line depreciation, or a fixed annual pay raise all form an AP, and \(S_n\) gives the running total. Note that interest-bearing accounts usually grow geometrically, not arithmetically — for those a compound interest calculator is the right tool.