Math 1 · Unit 9 · Lesson 3 · part 2 of 5

Multiplying by the same number: the common ratio

Adding gives arithmetic, multiplying gives geometric.

Multiplying by the same number: the common ratio

Adding gives arithmetic, multiplying gives geometric. This lesson is the multiplying one. A geometric sequence is a list that multiplies by the same number to get each next term. That number is the common ratio, written r.

Find r: divide

Take any term and divide it by the one before it.

2, 8, 32, 128: 8 ÷ 2 = 4, and 32 ÷ 8 = 4. r = 4.

3, 6, 12, 24: r = 2.

r can be a fraction

400, 200, 100, 50: 200 ÷ 400 = 12. r = 12.

81, 27, 9, 3: r = 13.

Write a fraction ratio as a stacked fraction.

Grow or shrink?

r bigger than 1: the terms grow.

r between 0 and 1: the terms shrink.

The same growth and decay rule as Unit 7, Lesson 3, because multiplying by more than 1 makes things bigger and by less than 1 makes them smaller.

The same two kinds of rule as Lesson 2, with multiplying in place of adding. A nursery plants 2, 8, 32, 128 trees in years 1, 2, 3, 4.

Recursive rule: the stairsa₁ = 2, a_n = 4 · a_{n − 1}

Say: a sub 1 is 2, and a sub n is 4 times a sub n minus 1. Each year is 4 times the year before.

Explicit rule: the elevatora_n = 2 · 4^{n − 1}

Say: a sub n is 2 times 4 to the n minus 1. Straight to any year.

a_n = a₁ · r^{n − 1}

Why n − 1 again?The same reason as Lesson 2. The first term has not been multiplied yet: it is where you start. Reaching the nth term takes n − 1 multiplications. Year 6 at the nursery: 2 · 4⁵ = 2 · 1024 = 2,048 trees. Power first, then times 2.

Where to go next

Next: It is an exponential: the ratio is the growth factorThe dots sit on an exponential curve, so a geometric sequence is the exponential family from Unit 7 in... All the partsGeometric Sequences, Exponentials Made of Dots