— GUIDE —

HOW TO SOLVE NUMBER SEQUENCES FASTER

When a number sequence looks opaque, check these five pattern families in order.

UPDATED 2026-08-12 · 5 MIN READ · BY THE VARIANCE TEAM

1. Write the differences

Start by subtracting adjacent terms. For 4, 7, 13, 22, 34, the differences are 3, 6, 9, 12. They increase by 3, so the next difference is 15 and the next term is 49. If first differences do not help, check the differences between the differences.

2. Check ratios and multiply-then-adjust rules

For 3, 7, 15, 31, 63, each term is the previous term doubled, then increased by 1. The next is 63 × 2 + 1 = 127.

3. Split alternating terms

If one rule almost works, inspect the odd and even positions separately. In 2, 10, 4, 20, 8, 40, the odd-position sequence is 2, 4, 8 while the even-position sequence is 10, 20, 40. The next term is 16.

4. Combine earlier terms

Test whether each term uses the previous two. In 2, 5, 7, 12, 19, 31, every term is the sum of the two before it, so the next term is 19 + 31 = 50.

5. Recognize familiar families and digit tricks

Memorize squares, cubes, powers of 2, primes, and factorials. If the arithmetic patterns fail, inspect the digits: they may be reversed, concatenated, or summed. For 1, 4, 9, 61, 52, 63, the terms are reversed squares: 1, 4, 9, 16, 25, 36. The next is reversed 49, or 94.

The 30-second checklist

  1. Differences, then second differences.
  2. Ratios, then multiply-and-adjust.
  3. Odd and even positions separately.
  4. Previous-two-term recurrences.
  5. Known number families and digit operations.

Give each check a few seconds, then move on. Staring at the same terms rarely helps. Try the examples in our practice set under time pressure.