What it does
In an arithmetic progression (AP) each term is the one before plus the same number, the common difference d: 3, 7, 11, 15, … has d = 4. Drawn as bars, an AP is a staircase with equal steps. This page draws that staircase and works out, with your numbers:
- the nth term aₙ = a + (n − 1)d,
- the sum of the first n terms Sₙ = n/2 [2a + (n − 1)d],
- the same sum by Gauss’s pairing: write the list forwards and backwards; every pair adds to first + last, so the sum is n/2 × (first + last),
- and which term a given number is (or that it is not a term at all).
It follows the Class 10 chapter arithmetic progressions.
Every term of an AP lies on a straight line, like the lines in the linear pairs; the sum Sₙ is a quadratic in n, as in the quadratic explorer. For patterns in whole numbers more generally, see the number theory toolkit.
How to use it
- Set the first term a, the common difference d and the number of terms n with the sliders or boxes (n up to 500 in the box).
- Read the first few terms at the top and see them as bars. Negative terms hang below the line in a different colour.
- Tick Show Gauss’s pairing to stack an upside-down copy of the staircase on top. Every column then has the same height, first + last, which is why the sum is n × (first + last) / 2.
- Type a number in Which term is to find its position.
- The quick buttons give a basic AP, the numbers 1 to 100, a decreasing AP and a savings plan.
A worked example
The page starts with 3, 7, 11, … (a = 3, d = 4) and n = 10.
- 10th term: 3 + 9 × 4 = 39.
- Sum: 10/2 × [6 + 9 × 4] = 5 × 42 = 210.
- Gauss: first + last = 3 + 39 = 42, and 10 terms make 5 pairs of 42, again 210.
- Which term is 39? 39 = 3 + (n − 1) × 4 gives n = 10.
Savings plan: save ₹500 in month 1 and ₹100 more each month. After 12 months the 12th deposit is ₹1 600 and the total is 12/2 × (500 + 1 600) = ₹12 600.
Tips
- If you know two terms, subtract them to find d: the 10th term minus the 5th term is 5d.
- A term number must be a whole number of 1 or more; if the formula gives anything else, the value is not in the progression.
Check yourself
Four fresh questions each time: an nth term, a sum, which term a number is, and 1 + 2 + … + m. Type the answer, press Check, and open the worked answer if you need it. Nothing is saved.
FAQ
Why does Gauss’s picture switch off for negative terms? Stacking bars only makes a clean rectangle when they all stand on the line. The formulas work for any AP.
Is this from the NCERT book? It covers the same topic with fibub’s own examples and pictures. fibub is not part of NCERT.
Where the numbers come from
- Standard results for arithmetic progressions: aₙ = a + (n − 1)d and Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), with the classic story of Gauss adding 1 to 100. Written for fibub.
- Checked against direct addition of the terms for 500 random progressions and against Python's sum(range(1, 101)) = 5050.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.