Sequences — missing number

What this practises

In these questions a rule links one number to another, and one number is hidden behind a question mark. You work out the rule from the numbers you can see, then use it to fill the gap.

This practises finding a rule from a few examples and then checking it, 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: finding it is the whole skill, and the steps below make it easier.

How to go about it

Some questions show pairs, like the input and output of a machine. Others show a small grid where every row follows the same rule. The steps are the same for both.

1. Find a complete pair or row, one with no question mark.
2. Try comparing how each number changes into the next.
3. Make a guess at the rule, then test it on a second complete pair or row.
4. Only when it works every time, use it on the question mark.

Ask yourself: does my rule work for every pair or row I can see?

Worked example: pairs

(2, 15); (3, 22); (4, 29); (5, ?)

The right-hand number goes up by 7 each time the left-hand number goes up by 1, so the rule starts with × 7. Test it on one whole pair: 2 × 7 = 14, and 14 + 1 = 15. The rule is × 7 then + 1. Check it on another pair: 3 × 7 = 21, and 21 + 1 = 22, which matches.

Now use it on the last pair: 5 × 7 + 1 = 36.

Worked example: grids

• 2, 5, 11
• 4, 9, 19
• 1, 3, ?

Take a row with no question mark and see how each number turns into the next. In the first row, 2 becomes 5 and 5 becomes 11: double it, then add 1. The second row agrees (4, 9, 19), so apply the same rule to the last row: 1, 3, 7.

A machine turns the first number of each pair into the second, always using the same rule.
(3, 24); (4, 29); (5, 34); (6, 39); (7, ?)
What does the machine give for 7?