The chapter opens with a person's ancestors doubling each generation: 2, 4, 8, 16 … This is a geometric progression growing in front of you. Drag the slider to grow the family tree, then explore every other idea in the chapter in the tabs below.
A sequence is a function whose domain is the natural numbers — plug in n, get the term aₙ. Pick a rule from the textbook, or write your own, and watch the terms appear one position at a time.
This sequence's domain is all of ℕ — it never ends. We're only ever looking at a window of it.
A series is the indicated sum a₁ + a₂ + a₃ + … + aₙ, written compactly with sigma notation Σₖ₌₁ⁿ aₖ. Set the bounds and the rule for the term, and watch each term stack onto the running total.
"Series" refers to the indicated sum, not the total itself. 1 + 3 + 5 + 7 is a finite series with four terms; its sum is 16.
In a G.P., every term after the first is the previous term times a fixed common ratio r: a, ar, ar², ar³, … The general term is aₙ = arn−1. Drag a and r and watch the bars grow, shrink, or flip sign.
|r| > 1 → terms grow without bound.
The sum of the first n terms is Sₙ = a(rⁿ − 1)/(r − 1) (for r ≠ 1). When |r| < 1 and n → ∞, the terms shrink to nothing and the running sum converges to a finite limit S∞ = a/(1 − r).
Each bar is the running total Sₖ for k = 1 … n. The dashed line is the infinite-sum limit a/(1−r), only meaningful when |r| < 1.
For positive numbers a, b: A = (a+b)/2 and G = √(ab). Draw a semicircle on a diameter of length a+b, split at the point between a and b, and the perpendicular chord there has length exactly G — while the radius is exactly A. That's a visual proof that A ≥ G, always.
Radius (gold) = A. Perpendicular chord at the split point (mint) = G. The gold radius never gets shorter than the mint chord — equal only when a = b.
Not every sequence has a closed formula. Fibonacci's rule looks back two steps: a₁=a₂=1, aₙ = aₙ₋₁ + aₙ₋₂. Watch the ratio of consecutive terms settle toward the golden ratio φ ≈ 1.618, and see the squares spiral outward.
Three of the chapter's story problems, made adjustable.
A culture doubles every hour. Starting count and hours elapsed → population.
₹P compounded annually at rate r% behaves like a G.P. with ratio (1 + r/100).
Every generation back doubles the ancestor count — same G.P. as the hero above.
A person mails a letter to k friends; each copies it to k more. Cost per stamp given.