Geometric Sequences

Multiply, do not add

In an arithmetic sequence, each term is found by adding the same number. In 3, 6, 12, 24, each term is found by multiplying by the same number: 3 × 2 = 6, 6 × 2 = 12 and 12 × 2 = 24. A sequence that multiplies by the same number every time is a geometric sequence.

Because each term is double the one before, the gaps double too: 3, then 6, then 12. The terms grow faster and faster, because each rise is twice the rise before it.

The common ratio

To test whether a sequence is geometric, divide each term by the term before it. For 3, 6, 12, 24: 6 ÷ 3 = 2, 12 ÷ 6 = 2 and 24 ÷ 12 = 2. The answer is the same every time, so the sequence is geometric, and that number, 2, is its common ratio.

The common ratio is usually called r, and the first term is called a. Here a = 3 and r = 2. Subtracting neighbors tests for an arithmetic sequence, and dividing them tests for a geometric one. 3, 6, 12, 24 is not arithmetic, because its differences 3, 6 and 12 are not equal.

Each new block of dots is as big as the whole term before it, so each term is the one before it times 2.

Jumping to any term

Term 2 is the first term multiplied by the ratio once: 3 × 2. Term 3 has been multiplied twice: $3\times 2\times 2=3\times 2^2$3 × 2 × 2 = 3 × 2². Term 4 has been multiplied three times: $3\times 2^3=24$3 × 2³ = 24. Each term has been multiplied one time fewer than its position, because the first term has not been multiplied at all.

So term n is $3\times 2^{n-1}$3 × 2ⁿ⁻¹. In general, the nth term of a geometric sequence with first term a and common ratio r is $a\times r^{n-1}$a × rⁿ⁻¹. Term 8, for example, is $3\times 2^7=3\times 128=384$3 × 2⁷ = 3 × 128 = 384, found without writing out the terms in between.

The power of 2 is always one less than n. The first term is $3\times 2^0=3\times 1=3$3 × 2⁰ = 3 × 1 = 3.

A ratio less than 1

The common ratio can be less than 1. Start at 16 and multiply by $\frac{1}{2}$1/2 each time: 16, 8, 4, 2, 1, $\frac{1}{2}$1/2, $\frac{1}{4}$1/4, … The ratio is $8÷16=\frac{1}{2}$8 ÷ 16 = 1/2, and the terms shrink. They get closer and closer to 0, but no term ever reaches 0, because half of a number bigger than 0 is still bigger than 0.

The ratio can also be negative. With a = 1 and r = −2, the terms are 1, −2, 4, −8, 16, …, and the sign changes at every step, because multiplying by a negative number changes the sign.

Multiplying by $\frac{1}{2}$1/2 again and again: 16, 8, 4, 2, 1. Each jump is half as long as the one before, and the terms close in on 0.