Sequences that multiply by the same number each time, growth and decay, and surd ratios.
What you will learn
- Find the common ratio
- Continue and find later terms of a geometric sequence
- Model doubling and halving
Multiply by the same number
3, 6, 12, 24, ... multiplies by 2 each time: the common ratio is 2.
Find it by dividing a term by the one before: 24 ÷ 12 = 2.
The ratio can be a fraction (64, 32, 16, ... has ratio 1/2), negative (2, −6, 18, ...) or a surd (1, √2, 2, 2√2, ...).
The nth term is (first term) × ratioⁿ⁻¹, so the 10th term of 3, 6, 12, ... is 3 × 2⁹ = 1536.
Growth and decay work the same way. If something starts at 40 and doubles every day, after 5 days it is 40 × 2⁵ = 1280. A ball dropped from 640 cm that bounces to half its height each time reaches 640 × (1/2)³ = 80 cm on bounce 3.
A missing term between two known ones: in 3, □, 27, the jump from 3 to 27 is two multiplications, so ratio × ratio = 27 ÷ 3 = 9, the ratio is 3, and the missing term is 9.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- Multiply by the same number
- Try it: See growth
7 practice questions, marked as you go; about 20 minutes in all.
Key terms
Common ratio
In this unit: Sequences
Term-to-term rules, the nth term of linear sequences, special sequences, geometric sequences and (Higher) the nth term of quadratic sequences.