DWG NO. 09 — Unit 9 of 10
6 online lessons · 1 in-person Review & Practice Day
Back in Unit 1, a function took a real number in and gave a real number out. A sequence is what happens when you restrict the input to positive integers only — \(a_{1}\), \(a_{2}\), \(a_{3}\), … is really just f(1), f(2), f(3), … with a new name and notation. This unit studies the two simplest patterns a sequence can follow — constant difference (arithmetic) and constant ratio (geometric, the discrete cousin of the exponential functions from Unit 3) — and what happens when you add their terms up to form a series.
From there the unit turns to two tools that don't depend on any particular sequence: mathematical induction, a proof technique for statements about every positive integer at once, and the Binomial Theorem, which expands (a + b\()^{n}\) without multiplying it out by hand. The unit closes with counting — permutations, combinations, and the counting principles that quietly power both the Binomial Theorem's coefficients and a first look at probability.
Arithmetic Sequences and Series
Sequences with a constant common difference, their explicit formula, and the sum of a finite arithmetic series.
→Geometric Sequences and Series
Sequences with a constant common ratio, and the sum of finite and (when it converges) infinite geometric series.
→Summation Notation and Series Properties
Reading and writing sigma notation, its algebraic properties, and closed-form formulas for common sums.
→Mathematical Induction
Proving a statement true for every positive integer using a base case and an inductive step.
→The Binomial Theorem
Expanding (a + b\()^{n}\) using binomial coefficients and Pascal's Triangle, and finding a single term directly.
→Counting Principles
The fundamental counting principle, permutations, combinations, and using them to compute basic probabilities.
→Review & Practice Day 9 — bring your questions from lessons 9.1–9.6. You'll work practice problems, get feedback, and take the Unit 9 test.