Multiply by the same number
Geometric progression
Doubling squares
Another pattern of squares: the number of green squares is 3, 6, 12, 24, … At each stage the number of squares is doubled.
How do the terms 3, 6, 12, 24 grow?
What this lesson covers
The idea
A geometric progression (GP) is a sequence in which each term is the previous term multiplied by a constant, the common ratio r, so the ratios of consecutive terms are all equal.
Doubling squares
Another pattern of squares: the number of green squares is 3, 6, 12, 24, … At each stage the number of squares is doubled.
How do the terms 3, 6, 12, 24 grow?
- The same number is added each time
- Each term is the previous one multiplied by the same number
- There is no pattern
Is the ratio constant?
For each list, press Show the ratios (each term divided by the one before). Is the ratio always the same? Then it is a GP.
Constant ratio
A geometric progression (GP) is a sequence in which each term is the previous term multiplied by a constant, the common ratio r. So the ratios of consecutive terms are all equal.
3, 6, 12, 24, 48, 96, … : 6 ÷ 3 = 12 ÷ 6 = 24 ÷ 12 = 48 ÷ 24 = 96 ÷ 48 = 2. The common ratio is 2.
Notes
A geometric progression (GP) is a sequence in which each term is the previous term multiplied by a constant, the common ratio r, so the ratios of consecutive terms are all equal.
Check yourself
Which of these is a GP?
What is the common ratio of 2, −6, 18, −54, …?
Answer: -3
−6 ÷ 2 = −3, 18 ÷ (−6) = −3, −54 ÷ 18 = −3. The common ratio is −3.
A GP starts 4, 12, 36, … What is the next term?
Answer: 108
r = 12 ÷ 4 = 3, so the next term is 36 × 3 = 108.
Is 1, −1, 1, −1, 1, … a GP?
- 1, 4, 7, 10, …. The differences are 3, 3, 3. That is an AP. The ratios 4, 7/4, 10/7 are not equal.
- 2, 4, 6, 8, …. The ratios are 2, 3/2, 4/3, which are not equal. (This one is an AP.)
- 1, 3, 9, 27, 81, … — correct. Yes. Every term is 3 times the one before, so r = 3.
- Yes, with common ratio −1 — correct. Yes. Each term is −1 times the one before: 1 × (−1) = −1, −1 × (−1) = 1.
- No, the terms do not grow. A GP does not have to grow. It only needs a constant ratio, and here the ratio is −1 every time.
- No, a ratio cannot be negative. A common ratio can be negative. The terms then change sign, as here.