What the question looks like
Number series questions open the Quantitative Skills section, and you will see a block of them. Each one shows a short list of numbers separated by commas and asks for a missing term. The wording is almost always the same:
Look at this series: 3, 7, 11, 15, ... What number should come next?
Most of the time the blank is at the end. Sometimes it sits in the middle, marked with a line, like 2, 4, ___, 16, 32. Either way, there are four answer choices and exactly one of them fits the rule.
The numbers are small on purpose. You will never need a calculator, and you should never need long division. If your arithmetic is getting ugly, you have probably guessed the wrong rule.
How to solve it
- Find the gaps. Write the difference between each pair of neighbors above the series. For 3, 7, 11, 15 the gaps are +4, +4, +4.
- If the gaps are all the same, you are done. Apply that gap once more to get the next term.
- If the gaps are not the same, look for a second-level pattern. Are the gaps growing by a fixed amount? Does the series multiply instead of add? Do two different operations take turns, like times 2 then minus 1?
- If nothing fits, check for two interleaved series. Read every other number: the 1st, 3rd, 5th terms form one series and the 2nd, 4th, 6th terms form another.
- Apply your rule to produce the missing term, then confirm it appears among the choices. If it does not, your rule is wrong; go back to step 3.
The whole process should take about 35 seconds. The most common series are add-a-constant, subtract-a-constant, multiply-by-a-constant, alternating operations, and interleaved pairs. Once you have seen each type a few times, you will recognize it on sight.
Worked examples
Example 1
Look at this series: 6, 13, 20, 27, ... What number should come next?
- A.34
- B.33
- C.35
- D.36
Show answer
Answer: A: 34
The gaps are +7, +7, +7, so the next term is 27 + 7 = 34. Choice B, 33, comes from adding 6 instead of 7, which happens when you compute one gap in your head and get it slightly wrong. Write the gaps down so you can see they match.
Example 2
Look at this series: 2, 6, 18, 54, ... What number should come next?
- A.108
- B.162
- C.150
- D.216
Show answer
Answer: B: 162
The gaps are +4, +12, +36, which are not equal, so try multiplying: 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54. Each term is 3 times the one before, so the next is 54 × 3 = 162. Choice A, 108, is what you get by doubling 54, a natural mistake if you only glance at the last two terms.
Example 3
Look at this series: 5, 10, 8, 16, 14, 28, ... What number should come next?
- A.24
- B.30
- C.26
- D.56
Show answer
Answer: C: 26
The series goes up, down, up, down, which signals alternating operations. Check: 5 × 2 = 10, 10 − 2 = 8, 8 × 2 = 16, 16 − 2 = 14, 14 × 2 = 28. The last step was a multiply, so the next step is a subtract: 28 − 2 = 26. Choice D, 56, is the trap for anyone who multiplies twice in a row.
Example 4
Look at this series: 40, 3, 36, 6, 32, ___, 28, 12. What number should fill the blank?
- A.8
- B.10
- C.24
- D.9
Show answer
Answer: D: 9
The numbers jump between big and small, so read every other term. The odd positions give 40, 36, 32, 28, which drop by 4 each time. The even positions give 3, 6, ___, 12, which rise by 3 each time, so the blank is 6 + 3 = 9. Choice A, 8, comes from assuming the small series doubles, but 6 doubled would be 12 and the next term would then have to be 24, not 12.
Less common series
About one series in four uses a format other than plain numbers. None of these is harder than the ordinary kind once you know the move, and each shows up on most forms of the test.
Letters mixed with numbers
A term like C4 or 12K is two series wearing one coat. Split it: solve the number part first, because numbers narrow the choices faster, then check the letters only if two choices are still alive. In A2, C4, E6, G8, the numbers climb by 2 and the letters skip one each time, so the next term is I10. If you are not sure of a letter's position, write the alphabet in the margin once at the start of the section and count on it. A plus sign or a times sign printed inside a term, as in 5 + A, is decoration, not arithmetic; treat the term as a pair.
Letters only
Letters behave like numbers 1 through 26. A, C, F, J skips 1, then 2, then 3, so the next skip is 4 and the term is O. Going backward through the alphabet is just subtraction: Z, W, T, Q drops three each time, so N comes next.
Roman numerals
Write the ordinary number above each numeral before you look for a pattern, then solve, then convert the answer back. II, IV, VI, VIII is 2, 4, 6, 8, so the next is 10, which is X. You need only I, V, and X for the test: a smaller symbol after a larger one adds, a smaller one before a larger one subtracts, so IV is 4 and VI is 6. Expect one or two of these per test at most.
Squares and square roots
A series that jumps between big and small by a lot, like 64, 8, 49, 7, 36, is alternating between a perfect square and its root. Know the squares from 2 to 12 by heart. A series such as 1, 4, 9, 16 is the squares themselves, and 2, 4, 3, 9, 4, 16 pairs each number with its square.
Decimals and fractions
Do not let the form distract you. 0.5, 1, 2, 4 doubles, exactly as 1, 2, 4, 8 does. If a decimal at the start is confusing, find the rule from the whole numbers later in the series and apply it to the last term. For fractions, look for a fixed step: 1/3, 2/3, 1, 4/3 adds one third each time, so the next is 5/3. Rewriting 1 as 3/3 makes the step visible.
One number is wrong
Occasionally the stem asks which term breaks the pattern instead of what comes next. Read the stem before you solve, or you will hunt for a next term that is not being asked for. Find the rule that fits all the terms but one; the odd one out is the answer. In 5, 10, 15, 21, 25, 30 every step is 5 except the one that lands on 21.
Next two terms
When the stem asks for the next two numbers, the choices are pairs. This usually signals an alternating or growing-difference series where a single next term would not prove you found the rule. Work both steps and check that the pair matches a choice exactly.
Traps to watch for
- Checking only one gap. A series like 4, 8, 12, 24 looks like add 4 until you reach the last pair. Always compute every gap before you commit to a rule.
- Missing the direction change. If a series rises and falls, it is almost certainly alternating operations or two interleaved series. Do not force a single rule onto it.
- Blank in the middle. When the blank is not at the end, use the terms on both sides of it. The term after the blank is a free check on your answer.
- Picking a choice that looks close. The choices are designed so that each common mistake lands on a wrong option. If your number is not in the list, do not pick the nearest one; recheck the rule.
Pacing
You have about 35 seconds per question across the whole Quantitative Skills section, and number series should run a little faster than that so you can bank time for the comparison questions. If you have tried two rules and neither matches a choice, pick the most reasonable option and move on. There is no guessing penalty on the HSPT, so a blank is always worse than a guess.
Practice
Try a timed set of number series and every other Quantitative question type in our free diagnostic.
Shaky on the skill itself? Start with the lessons on decimals and exponents and roots, then come back to the question format.