CCAT number series questions, and the checklist that cracks them

What a number series question looks like

A number series question shows a short run of numbers and a blank. "What comes next? 5, 7, 11, 17, 25, ?" You pick one of five answers. There is no story to read and no unit to convert. That makes it one of the fastest questions on the CCAT when you see the rule, and one of the worst time sinks when you don't.

The test gives you 50 questions in 15 minutes. That is about 18 seconds a question. A number series should take less than that. If you are still guessing at rules at 30 seconds, the problem is not arithmetic. It is the order you test rules in.

Two kinds show up in our bank. A plain sequence follows one rule from start to finish. An interleaved sequence is two sequences woven together: the 1st, 3rd and 5th terms follow one rule, and the 2nd, 4th and 6th follow another. Both live in the math domain. (Letter series, which use the same thinking on the alphabet, sit with the logic questions.)

The pattern checklist, in the order to try it

Work down this list. Most series fall to the first four checks. Stop at the first rule that fits every step, not just the first two.

1. Constant difference. Subtract each term from the next. 4, 11, 18, 25: the gap is 7 every time, so the next term is 32. This is the warm-up and it rarely appears at the harder levels.
2. Second differences. If the gaps are not equal, write the gaps down and look at them as their own sequence. 3, 5, 9, 15, 23 has gaps 2, 4, 6, 8. The gaps rise by 2, so the next gap is 10 and the answer is 33. Almost every medium-difficulty number sequence in our bank is this type.
3. Ratio. If the numbers grow fast, divide instead of subtract. 3, 6, 12, 24: each term doubles, so 48. Check one more pair before you commit; a ratio that only fits the first two terms is not the rule.
4. Multiply, then add or subtract. The harder version. 2, 7, 22, 67: each term is three times the last, plus 1, so the next is 3 × 67 + 1 = 202. The tell is in the gaps: 5, 15, 45. The gaps themselves triple. When the differences grow by the same ratio, the rule is "multiply by that ratio, then adjust", and you can finish from the gaps alone: the next gap is 135, and 67 + 135 = 202.
5. Interleaved. If the numbers jump up and down, or a big number sits next to a small one, split the series into odd and even positions and solve each half. More on this below.
6. Squares, cubes and primes. 1, 4, 9, 16, 25 are squares, so 36 comes next. 2, 5, 10, 17 are squares plus one, so 26. 2, 3, 5, 7, 11 are primes, so 13. Knowing the squares to 15 × 15 and the primes to 50 makes these instant.
7. Each term from the two before it. 2, 3, 5, 8, 13: each term is the sum of the previous two, so 21. If nothing above fits and the gaps look like the terms themselves, try this.

How to spot an interleaved sequence fast

Interleaved sequences are built to look random. Three signs give them away in the first two seconds:

• The series zigzags. It goes up, down, up, down. A single-rule sequence rarely changes direction.
• It is longer. Plain series in our bank show five terms. Interleaved ones show six, so each half gets three.
• Sizes alternate. A big number sits between two small ones, or the reverse.

When you see any of these, read every other term. Take 3, 20, 6, 17, 12, 14, ?. The odd positions are 3, 6, 12: they double. The even positions are 20, 17, 14: they drop by 3. The blank is the 7th term, an odd position, so it comes from the doubling strand: 24.

Now the trap. The even strand's next term would be 11, and in a question like this you will often find 11 among the choices. In our bank, 90 of the 136 interleaved questions list the next term of the wrong strand as an answer. Before you click, count positions: an odd-numbered blank belongs to the odd strand.

Traps that cost points

• Checking two terms instead of all of them. 2, 4, 8 could be doubling or could be gaps of 2 then 4. Confirm your rule on every step before you answer.
• Reusing the last gap. With gaps 6, 12, 18, 24, the tempting answer adds 24 again. The gaps are growing, so the next one is 30.
• Stopping at "it doubles". In a multiply-and-add series, plain doubling gets close. Our bank puts the near miss in the choices: for 3, 5, 11, 29, 83, one option is 166, which is just 83 doubled. The rule is "triple, then subtract 4", which gives 245.
• Solving the wrong strand. See above. Count positions before you answer.
• Arithmetic under time. Multiply-and-add answers can run into the hundreds and thousands. Do the last step twice. Our simulator has no calculator, so practice by hand.

How long to spend, and when to skip

Give a number series one pass through the checklist. If the first four checks fail and the series does not zigzag, you are looking at something unusual. Guess, flag it mentally, and move on. Spending 45 seconds on one question costs you two others you could have answered.

The reverse is also true: a number series you can read in five seconds banks time for the word problems and the tables, which take longer.

Eight practice questions with worked answers

These come from our bank, at medium and hard difficulty. Pick an answer, then open the explanation. The note under each one names the trap the wrong answers are built on.

Where number series fit in the CCAT

Number series are math questions. In our simulator, each 50-question test has 18 math questions spread across 12 math families, and two of those families are sequences: plain number sequences and interleaved ones. That is our simulator's mix, built to match the format; Criteria does not publish a per-type breakdown.

The math page covers the other ten families, from percentages to tables and charts. If you also get stuck on story questions, our word problems guide walks through the setups for rates, work, averages and percent change.

How to practice number series

Read the checklist once, then drill. Twenty series in a row is enough to make the order automatic. In the app, a focus drill on number sequences and interleaved sequences times every answer and shows the rule for each one. The AI coach then reads your misses and tells you which pattern keeps costing you.

Then put it under the clock. A free CCAT practice test mixes series in with the other 49 questions, which is where the timing pressure actually bites. For more question types, start at the CCAT practice questions hub or the CCAT math questions page.