14. Sequences & Series
CollegeA sequence is an ordered list of numbers following a rule; a series is their sum. This chapter covers arithmetic and geometric patterns, their sum formulas, and the surprising idea that an infinite sum can be finite.
Arithmetic, geometric and their sums
An arithmetic sequence adds a fixed common difference d each step; a geometric sequence multiplies by a fixed common ratio r. Their n-th terms:
Geometric: an = a1 · r^(n−1)
A series is the sum of a sequence. The partial sums have closed forms:
Geometric sum: Sn = a1(1 − rⁿ)/(1 − r)
When |r| < 1, the terms shrink and the infinite geometric series converges to a finite total:
Gauss's trick and the vanishing tail
The arithmetic sum formula comes from a famous shortcut. To add 1 + 2 + … + 100, pair the ends: 1 + 100 = 101, 2 + 99 = 101, and so on — 50 pairs each summing to 101, giving 5050. That is exactly (n/2)(first + last): average the endpoints, multiply by how many terms.
The infinite geometric result feels paradoxical — how can adding forever give a finite answer? Because each term is a fraction of the last, the leftover "tail" shrinks toward nothing. Walk half the remaining distance to a wall each step (½ + ¼ + ⅛ + …): you approach the wall as closely as you like, and the total distance is exactly 1. Convergence happens precisely when |r| < 1; otherwise the terms do not shrink and the sum runs away to infinity.
Worked example: two sums
Find the sum of the first 20 terms of 3, 7, 11, 15, …
- Here a₁ = 3, d = 4, n = 20. The 20th term: a₂₀ = 3 + 19(4) = 79.
- Sum: S₂₀ = (20/2)(3 + 79) = 10 × 82 = 820.
Evaluate 6 + 2 + 2/3 + 2/9 + …
- Ratio r = 2/6 = 1/3, and |r| < 1, so it converges.
- S∞ = a₁/(1 − r) = 6 / (1 − 1/3) = 6 / (2/3) = 9.
Test your understanding
Is 0.9999… (repeating) equal to 1?
Yes, exactly. It is the infinite geometric series 0.9 + 0.09 + 0.009 + … with a₁ = 0.9, r = 0.1. Summing: 0.9/(1 − 0.1) = 0.9/0.9 = 1. They are two names for the same number, not "almost equal."
A ball dropped from 10 m rebounds to 60% of its height each bounce. What total vertical distance does it travel?
It falls 10 m, then each subsequent up-and-down is a geometric pair. Up-distances: 6 + 3.6 + … sum to 6/(1−0.6) = 15; the downs mirror them for another 15, but the very first drop of 10 is only counted once. Total = 10 + 2(15) = 40 m.
Why must |r| < 1 for an infinite geometric series to converge?
The formula contains rⁿ. If |r| < 1, then rⁿ → 0 as n → ∞, so Sₙ → a₁/(1 − r). If |r| ≥ 1, the terms do not shrink toward zero, the partial sums grow without bound (or oscillate), and no finite limit exists.
The sum 1 + 1/2 + 1/3 + 1/4 + … — does it converge?
No — this is the harmonic series, and it diverges to infinity, even though its terms shrink to zero. Terms shrinking is necessary but not sufficient for convergence. Grouping shows why: 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on — infinitely many chunks each exceeding ½.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Find the sum 1 + 2 + 3 + ... + 100.
- Use n/2 times (first + last)
- 100/2 x (1 + 100) = 50 x 101
- Sum = 5050
Example 2Find the infinite sum 1 + 1/2 + 1/4 + ...
- This is geometric with a = 1, r = 1/2
- Sum = a/(1 - r) = 1/(1 - 1/2)
- Sum = 2
Now you try
Work each one out first, then tap to reveal the worked answer.