Level 6: Geometric sequences

Patterns and sequences · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

In level 5 you added the same amount each time. In this level you multiply by the same amount each time, and the patterns behave completely differently: they explode (rice on a chessboard, paper folded to the Moon), shrink towards zero, or flip sign back and forth. You learn the formula for any term, how to tell arithmetic from geometric from neither, how to add up a geometric sequence with one clever subtraction, how to work with rules that build each term from the ones before, and how to choose the right kind of sequence for a real situation.

Lesson	You will be able to
Geometric sequences	Find the common ratio, use \(t_n = ar^{n-1}\), and handle halving and negative ratios
Why geometric sequences get huge	Estimate powers of 2 with \(2^{10} \approx 1000\), and explain the chessboard and paper-folding stories
Finding a and r	Rebuild a geometric sequence from two terms, and test whether a list is arithmetic, geometric or neither
Adding a geometric sequence	Prove and use \(S_n = \dfrac{a(r^n - 1)}{r - 1}\) by "multiply by \(r\) and subtract", and count every grain on the chessboard
Recursive sequences	Compute terms of rules like \(t_n = 3t_{n-1} - 1\) in a table or spreadsheet, find where they settle, and turn some into position rules
Arithmetic or geometric in the world	Choose and use the right model for bouncing balls, phone plans, bacteria, interest and radioactive fuel

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