What it does
In a geometric progression (GP) each term is the one before multiplied by the same number r, the common ratio: 8, 4, 2, 1, … has r = ½. This page draws a GP so you can see what the formulas say:
- Terms as bars. Change the first term a and the ratio r and the bars change at once. A negative r makes them flip up and down; r above 1 makes them grow fast.
- The running sum Sₙ is the black line. When |r| < 1 the terms shrink to nothing and the line levels off at the dashed sum to infinity, a/(1 − r). When |r| ≥ 1 there is no such line.
- AM ≥ GM. Two numbers a and b sit end to end as the diameter of a semicircle. The radius is their arithmetic mean (a + b)/2 and the half-chord standing at the join is their geometric mean √(ab). A half-chord can never be longer than a radius, so AM ≥ GM, with equality only when a = b. Tick the box to add the harmonic mean, with GM² = AM × HM.
- Money as a GP. Compound interest multiplies a sum by 1 + i/100 every year, and depreciation by 1 − i/100: both are GPs.
It follows the Class 11 chapter sequences and series (GP, sum to infinity, AM and GM) and the compound interest work from Class 8.
Arithmetic progressions, where you add instead of multiply, are in arithmetic progression staircase; the idea of a sum settling on a value is the limit in limit zoom, and binomial expansion comes next in the same book.
How to use it
- In Terms and sums, set a, r and the number of terms n (up to 15) with the sliders, or type values in the boxes.
- Read the list of terms, the nth term, the sum of n terms and the sum to infinity, all with your numbers.
- In AM ≥ GM, set two positive numbers and watch the radius (AM, blue) and the half-chord (GM, green).
- In Money as a GP, type a starting amount, a yearly rate and a number of years, and pick compound interest or depreciation.
A worked example
The page starts with a = 8, r = 0.5 and n = 8.
- Terms: 8, 4, 2, 1, 0.5, 0.25, 0.125, 0.0625.
- S₈ = 8 × (0.5⁸ − 1)/(0.5 − 1) = 15.9375.
- S∞ = 8/(1 − 0.5) = 16. After 8 terms the sum is still 0.0625 short of 16, exactly the size of the last term.
With a = 4 and b = 9: AM = 6.5, GM = √36 = 6 and HM = 72/13 ≈ 5.5385; 6² = 6.5 × 5.5385.
10,000 at 8 % a year for 10 years becomes 10,000 × 1.08¹⁰ ≈ 21,589.25. It first doubles after 10 whole years (the exact time is 9.006 years; the rule of 72 says 9).
Tips
- To find r, divide any term by the one before it.
- For 0 < r < 1 use Sₙ = a(1 − rⁿ)/(1 − r): it is the same formula with both signs flipped, and avoids negatives.
- A repeating decimal like 0.333… is a GP: 3/10 + 3/100 + … with r = 1/10, so it sums to 1/3.
Check yourself
Five fresh questions each time: a term, a sum of n terms, a sum to infinity, a geometric mean and a depreciation. Fractions like 16/3 are accepted. Nothing is saved.
FAQ
Why does the money part say teaching model? Real savings and loans have tax, fees and interest added monthly or daily. Here interest is added once a year at a fixed rate so the GP is easy to see.
Is this from the NCERT book? It covers the same topics with fibub’s own examples. fibub is not part of NCERT.
Where the numbers come from
- Standard results on geometric progressions: aₙ = a rⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1), S∞ = a/(1 − r) for |r| < 1, AM ≥ GM ≥ HM with GM² = AM × HM for two positive numbers, and compound growth P(1 + i/100)ⁿ. Written for fibub.
- Checked against term-by-term sums in Python and on 300 random progressions; the semicircle construction for the means is the classical one (the half-chord at the join is √(ab)).
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.