Unit 13
Sequences and series.
Arithmetic and geometric sequences, their sums, endless sums with a finite total, and Fibonacci in a sunflower.
Before this unit
The unit, lesson by lesson
Lessons
4 lessons · 20 practice questions · about an hour
Arithmetic sequences
A sequence that adds the same number at every step: the general term, how to find the difference, how Gauss added the numbers from 1 to 100 in pairs, and why the odd numbers always add up to a square.
Geometric sequences
A sequence that multiplies by the same number at every step: the general term, how to find the ratio, the sum of the first n terms, and why in the end it overtakes any arithmetic sequence.
Infinite series
What happens when you add terms forever: why a half plus a quarter plus an eighth adds up to exactly 2, the formula for the sum of a convergent geometric series, when there is no finite sum, and why 0.999… equals 1.
Fibonacci spirals
Why you count 8 and 13 or 34 and 55 spirals on a sunflower or a pine cone: a fixed turn between seeds, fractions of a turn that make spokes, the golden angle that lies near no simple fraction, and the Fibonacci numbers that close in on it.