Sequences — find the rule
A run of numbers was built by one rule; choose the written rule that fits every step.
What this practises
In these questions you are shown the start of a run of numbers and asked which rule built it. The answer is a sentence, not a number, and every option is a rule that sounds sensible.
This practises working out how a run is built and then testing a rule against every step, which is the kind of reasoning this section is about. The questions are at the Building level, the middle of the three levels used across this section. Don't worry if the rule isn't obvious at first: you can tell the options apart by testing them.
How to go about it
- Write the gaps between neighbouring numbers. Try comparing how each number changes into the next.
- Look at what the gaps are doing. Are they the same, do they grow by the same amount each time, or is each number the previous one times the same number? Some rules take two steps in turn and repeat, such as "multiply by 4, then add 3, and repeat" (3, 12, 15, 60, 63, …).
- Test the option you like against every step, not only the first.
- Cross out any option that fails once.
Ask yourself: does my rule work for every step in the run?
Worked example
4, 5, 8, 13, 20, …
The gaps are 1, 3, 5 and 7: each gap is 2 bigger than the one before. So the rule is "the gaps between the terms increase by 2".
The wrong options usually fit the first step and break later. "Each term increases by 1" fits the step from 4 to 5 and fails at the step from 5 to 8, so cross it out.
How to attack this question type
- 1
Write the gaps between neighbours before you read the options. The gaps usually point straight at the rule.
- 2
If the gaps are not steady, look at whether each term is the previous one times the same number, or whether the gaps themselves change by the same amount each time.
- 3
Cross out an option the moment it fails at any step. Only one rule fits the whole run.
Practice · untimed
These terms were built with one rule.
5, 15, 17, 51, 53, …
Which rule was used?
More in Numerical Reasoning
Number sequences
Find the missing term by working out the rule that generates the sequence.
Ratio and proportion
Share a quantity in a given ratio, or scale one quantity against another.
Rates and work
Scale a rate — workers, speed, cost per unit — up or down to a new quantity.
Age problems
Relate two ages now and at another time, then solve for one of them.
Sequences — missing number
Pairs or a small grid follow one rule; work the rule out and fill in the hidden number.
Sequences — odd one out
Every number is shown and one of them breaks the pattern the rest follow; find it.