Sequence Sums: What This Calculator Helps You Compute
A sequence is an ordered list of numbers (called terms) such as 3, 5, 7, 9, … . A series is what you get when you add some (or all) of those terms. So when people say “sequence sum,” they usually mean “the sum of the first n terms,” also known as a partial sum.
This sequence sum calculator focuses on the most common series types you’ll see in math, science, finance, and programming: arithmetic series (constant difference), geometric series (constant ratio), sums of consecutive integers (an arithmetic series with step 1), and classic power sums like 1 + 2 + … + n, 1² + 2² + … + n², and more. It also includes a partial sum table mode so you can see every term and the running total—useful for verification, teaching, and spotting patterns.
Arithmetic Series: Constant Differences, Fast Sums
An arithmetic sequence changes by adding the same amount each time. That amount is the common difference d. If the first term is a₁, then the sequence looks like:
The sum of the first n terms of an arithmetic sequence is called an arithmetic series. The reason arithmetic sums are so convenient is that they have a simple closed form:
Another equivalent form is:
That second version is especially helpful when you know the last term aₙ but not the difference d. This calculator supports both input styles: (a₁, d, n) or (a₁, aₙ, n). Under the hood, if you provide a₁ and aₙ, the calculator derives the difference using:
Arithmetic series show up everywhere: the total number of items added in equal increments, stair-step patterns, uniform installment changes, and many “sum from A to B” questions in algebra.
Why “Average of the Ends” Works
A classic insight is that an arithmetic series can be summed by pairing the first and last terms: (a₁ + aₙ), (a₂ + aₙ₋₁), and so on. Each pair totals the same value because the sequence increases evenly. There are n terms, so there are n/2 pairs (or the middle term stands alone when n is odd), which leads directly to: Sₙ = n(a₁ + aₙ)/2. The calculator displays the average term for this reason: the average of an arithmetic sequence is simply (a₁ + aₙ)/2.
Geometric Series: Constant Ratios and Exponential Growth
A geometric sequence changes by multiplying by the same ratio each time. That constant multiplier is the common ratio r. If the first term is a₁, then:
The sum of the first n terms is:
If r = 1, every term is a₁ and the sum is just Sₙ = a₁n.
Infinite Geometric Series (When It Converges)
One of the most useful properties of geometric series is the infinite sum. If the ratio shrinks in magnitude (meaning |r| < 1), then the terms get smaller and smaller and the series converges to a finite limit:
This is essential in modeling repeating-decimal expansions, decay processes, and “approach a limit” situations. The calculator can display the infinite sum when the ratio qualifies, and it will clearly note when it does not.
Consecutive Integer Sums: From m to n (With Optional Step)
Many real problems ask for the sum of consecutive integers: 1 + 2 + … + n, or 25 + 26 + … + 100. These are arithmetic sequences with difference 1. This calculator generalizes it slightly by letting you choose a step. When step = 1, you’re summing every integer. When step = 2, you’re summing every other integer (evens or odds depending on start).
The key is to compute how many terms you actually have, then apply the arithmetic series formula using the first and last terms. In this mode the calculator reports:
- Count of terms (how many values are included)
- Last term used (after applying the step)
- Average of ends and the resulting sum
Power Sums: Closed Forms for 1^p + 2^p + … + n^p
Power sums are extremely common in algebra, calculus (Riemann sums), statistics (moments), and algorithm analysis. Instead of looping from 1 to n, mathematicians have derived closed forms. This calculator includes p = 1 through p = 4:
| Power | Sum | Common use |
|---|---|---|
| p = 1 | S = n(n+1)/2 | Totals, counting, triangular numbers |
| p = 2 | S = n(n+1)(2n+1)/6 | Variance/math proofs, geometry |
| p = 3 | S = [n(n+1)/2]² | Cube identities, algebra patterns |
| p = 4 | S = n(n+1)(2n+1)(3n²+3n−1)/30 | Higher moments, advanced sums |
Power sums grow quickly. For example, the p = 1 sum grows on the order of n², p = 2 grows ~ n³, p = 3 grows ~ n⁴, and so on. That growth rate is why closed forms matter: they give instant results even for large n.
Partial Sums and Tables: Seeing the Pattern Term by Term
A partial sum Sₖ is the sum of the first k terms. Seeing S₁, S₂, S₃, … is often the fastest way to validate your model, check your arithmetic, or understand whether the series is growing linearly, quadratically, or exponentially.
The table mode in this calculator outputs three columns:
- k: the term index
- aₖ: the term value
- Sₖ: the cumulative sum up to that term
For arithmetic sequences, the running total increases by steadily larger chunks because each new term is bigger than the previous. For geometric sequences, the cumulative sum can either explode (|r| > 1), oscillate (negative r), or approach a limit (|r| < 1). Using the table helps you see this behavior immediately.
Practical Tips: Avoiding Common Mistakes
- Don’t confuse n with the last term. n is the number of terms, while aₙ is the value of the nth term.
- Check sign and ratio carefully. A negative ratio in a geometric series alternates signs: +, −, +, −, …
- Use the right formula for r = 1. The geometric sum formula divides by (1−r); when r = 1 you must use Sₙ = a₁n.
- Mind rounding for decimals. Very large exponents and long tables can show tiny floating-point rounding differences.
- Use table mode to verify. If a closed-form result surprises you, generate a few rows and compare.
FAQ
Sequence Sum Calculator – FAQs
Quick answers about arithmetic and geometric series, partial sums, power sums, and rounding.
A sequence sum (or series sum) is the total you get when you add the terms of a sequence. For example, adding the first n terms of an arithmetic sequence gives an arithmetic series sum.
A sequence is an ordered list of numbers (terms). A series is the result of adding some or all of those terms.
If the first term is a₁, the common difference is d, and there are n terms, the sum is Sₙ = n/2 × (2a₁ + (n−1)d).
If the first term is a₁, the common ratio is r, and there are n terms, the sum is Sₙ = a₁(1−rⁿ)/(1−r) when r ≠ 1. If r = 1, then Sₙ = a₁n.
An infinite geometric series converges only when |r| < 1. In that case, the sum is S∞ = a₁/(1−r).
Count the terms: k = n−m+1, then sum = k(m+n)/2 (works for integers and evenly spaced values).
Power sums have closed forms. For example, 1+…+n = n(n+1)/2 and 1²+…+n² = n(n+1)(2n+1)/6.
Some calculations use decimals and powers that rely on floating-point arithmetic in the browser, which may introduce tiny rounding differences for large n or extreme ratios.
It’s similar in structure (term-by-term rows), but it’s for sequences: each row shows the term value and the running (cumulative) sum.
Yes—use the formulas and step-by-step explanations to show your method. Always follow your course notation and rounding rules.