What you will learn in Sequences & Series

1. 1. Patterns, Sequences and the nth Term

   From a growing picture to a rule that generates it — and why three terms never settle which rule it is.

2. 2. Arithmetic Progressions

   A constant difference, which makes the nth term a straight line in disguise.

3. 3. Summing an AP

   Two staircases make a rectangle, and the sum falls out of counting it.

4. 4. Geometric Progressions

   A constant ratio. Multiplying beats adding eventually, whatever the numbers.

5. 5. Summing a GP

   Multiply by the ratio, subtract, and everything in the middle cancels.

6. 6. Infinite Geometric Series and Convergence

   Infinitely many pieces filling a finite square, and the two boundary ratios where it stops working.

7. 7. Arithmetic, Geometric and Harmonic Means

   Three averages, three lengths in one semicircle, and an inequality you can read off the picture.

8. 8. Special Series and Telescoping

   Sums of numbers, squares and cubes read off block arrangements — and a method where nearly everything cancels.

9. 9. Recursive Sequences and the Fibonacci Family

   A rule that refers to itself, and a ratio that settles on a number no fraction reaches.

10. 10. What Convergence Means (Informally)

    A sequence and a series are different objects. Naming which sentences here are the informal ones is the handoff to Limits & Continuity.