Module 2 - Finding the General Term
TL;DR
You need to be able to spot patterns in sequences of numbers and use this to find the next few terms in the sequence.
Common patterns are adding, subtracting, multiplying and dividing by a fixed number, add previous terms together, or patterns where the amount it increases by increases in a pattern.
Pattern Spotting
The first part of any investigation is to be able to spot patterns in sequences of numbers.
The most common types of patterns are:
increasing by a constant e.g. \(3, 7, 11, 15, 19, …\) where we increase by \(4\) each time
decreasing by a constant e.g. \(22, 15, 8, 1, -6, …\) where we decrease by \(7\) each time
multiplying by a constant e.g. \(2, 6, 18, 54, 162, …\) where we multiply by \(3\) each time
dividing by a constant e.g. \(128, 64, 32, 16, 8, …\) where we divide by \(2\) each time
adding together two previous terms e.g. \(2, 3, 5, 8, 13, 21, 34, …\) where each term is the sum of the previous two (that is \(5=2+3\), \(8=3+5\), \(13=5+8\) and so on)
the amount you increase by changes by a constant e.g. \(2, 3, 5, 8, 12, 17, 23, …\) where we add \(1\), then we add \(2\), then we add \(3\), then we add \(4\), and so on. There is a pattern to the numbers we add each time.
Find the next two terms for each of the following sequences:
- $$5, 8, 11, 14, \dots$$
- $$20, 17, 14, 11, \dots$$
- $$2, 4, 8, 16, \dots$$
- $$1, 1, 2, 3, 5, 8, \dots$$
- $$100, 90, 81, 73, \dots$$
- $$3, 9, 27, 81, \dots$$
- $$1, 4, 9, 16, 25, \dots$$
- $$0, 3, 8, 15, 24, \dots$$
- $$10, 20, 40, 80, \dots$$
- $$2, 5, 10, 17, 26, \dots$$
- $$1, 3, 4, 7, 11, 18, \dots$$
- $$80, 40, 20, 10, \dots$$
- $$2, 6, 12, 20, 30, \dots$$
- $$1, 8, 27, 64, \dots$$
- $$1, 3, 7, 15, 31, \dots$$
- $$2, 3, 5, 8, 12, 17, \dots$$
- $$100, 99, 97, 94, 90, \dots$$
- $$0.5, 1, 2, 4, 8, \dots$$
- $$2, 6, 18, 54, \dots$$
- $$1, 2, 6, 24, 120, \dots$$
Solutions
Sequence Practice
Spot the pattern and find the next two terms.