Building blocks

## Problem

Sort these infinite sequences into groups. (They are available on cards for cutting out.)

• How will you choose to define the groups?
• Are there some sequences which belong to more than one group?

Once you have sorted the sequences into groups, write a further sequence for each group.

How else could you have sorted the sequences?

$\dfrac12, -\dfrac{1}{4}, \dfrac18, -\dfrac{1}{16}, ...$

$3\sqrt2, 4\sqrt2, 5\sqrt2, 6\sqrt2,...$

$2,0,-2,-4,...$

$\ln 1, \ln 2, \ln 3, \ln 4, ...$

$-1, -\sqrt2, -2, -2\sqrt2, ...$

$1,1,1,1,...$

$1, \dfrac12, \dfrac13, \dfrac14, ...$

$-1, 1, -1, 1, ...$

$0.1, 0.01, 0.001, 0.0001, ...$

$3,2\dfrac13, 1\dfrac23, 1,...$

$4, 8, 16, 32, ...$

$\cos\left(\frac{\pi}{4}\right), \cos\left(\frac{2\pi}{4}\right), \cos \left(\frac{3\pi}{4}\right), \cos\left(\frac{4\pi}{4}\right), ...$

$1, 1\dfrac12, 1\dfrac23, 1\dfrac34, ...$

$100, 105, 110, 115, ...$

$3.75, 5.625, 8.4375, 12.65625, ...$