Read a number sequence from its gaps

The rule behind a sequence is often hard to see in the numbers themselves. It is usually easier to see in the gaps between them.

Write the difference between each pair of neighbours underneath. Here every gap is the same, so the sequence adds 3 each time and the next term is 19.

The gaps under 3, 4, 6, 9, 13 are 1, 2, 3 and 4. They are not equal, so no single amount is being added.

So do the same thing again, to the gaps. Each gap is 1 more than the one before it. The next gap is 5, and 13 + 5 = 18.

When the terms grow much faster than this, divide instead of subtracting. In 3, 6, 12, 24 the gaps are 3, 6 and 12 and keep changing, but every term is exactly twice the one before.

Gaps do not open every sequence. Some alternate between two rules, and a short sequence can have more than one sensible continuation. The practice tells you the rule it meant.

The gaps are already drawn under this sequence. See how they change, work out the gap that belongs in the empty place, then add it to 32.