Unit 9 of 10

Sequences, Series, and Induction

  • 8 lessons
  • Parts 201 to 216

A sequence is a list of numbers produced by a rule, and a series adds up its terms. This unit defines sequences explicitly and recursively, introduces factorials and sigma notation, and finds formulas for arithmetic and geometric sums.

Infinite geometric series show how adding infinitely many terms can give a finite total. The unit also covers proof by mathematical induction and the binomial theorem with Pascal’s triangle.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Sequences

  1. Introduction to Sequences and Factorials

    What a sequence is as a function of an integer index, how to read its terms from an explicit formula and where the counting starts, and how factorials work and simplify.

    Theory part 201Practice part 202

  2. Recursive Sequences and the Fibonacci Sequence

    How starting values and a rule that uses earlier terms define a sequence, how to generate its terms and the Fibonacci numbers, and why such a rule need not settle on a limit.

    Theory part 203Practice part 204

Series

  1. Sigma Notation and Partial Sums

    How to read and write sums in sigma notation, count their terms, split and shift them, and build the partial sums of a sequence, the running totals that give series their meaning.

    Theory part 205Practice part 206

  2. Arithmetic Sequences and Series

    How a constant difference gives a formula for any term of an arithmetic sequence, and how pairing the first and last terms adds up any number of its terms in one step.

    Theory part 207Practice part 208

  3. Geometric Sequences and Series

    How a constant ratio gives a formula for any term of a geometric sequence, and how multiplying a sum by that ratio and subtracting adds up any number of its terms in one step.

    Theory part 209Practice part 210

  4. Infinite Geometric Series and Convergence

    When adding all the terms of a geometric sequence gives a finite total, why the ratio must be less than 1 in absolute value, and how the sum formula turns repeating decimals into fractions.

    Theory part 211Practice part 212

Induction and the Binomial Theorem

  1. Mathematical Induction

    How to prove a statement for every integer from some starting point on by checking a first case and showing that each case implies the next, and why checking many cases is never a proof.

    Theory part 213Practice part 214

  2. The Binomial Theorem and Pascal’s Triangle

    How to expand any whole-number power of a sum with the binomial theorem, how Pascal's triangle, the factorial formula and counting all give its coefficients, and why the expansion stops.

    Theory part 215Practice part 216