HSPT: Number series: find the constant difference
Learn to spot a series that rises or falls by the same amount each step, and to continue it or fill a gap in it.
How do you find the next number when a series goes up or down by the same amount each time?
Short answer
Subtract each term from the next one. If every gap is the same, add that gap once to the last term.
Know first: Subtracting whole numbers, Negative numbers
Step 1 of 5
Subtract neighbors to find the step
In a constant-difference series, you get each term by adding or subtracting the same number every time. That number is the common difference.
To find it, subtract each term from the one after it and write the result between them. If every difference matches, you have the rule. Add it once more to the last term.
| Series | Gaps | Next term |
|---|---|---|
| 12, 21, 30, 39 | +9, +9, +9 | 48 |
| 85, 78, 71, 64 | −7, −7, −7 | 57 |
| 8, 22, 36, 50 | +14, +14, +14 | 64 |
The gap can be any size, and it can be negative.
Step 2 of 5
Check every gap, then take one step
Worked example: A rising series
Which number continues the pattern? 14, 26, 38, 50, 62, …
- A62
- B74
- C86
- D72
- 1
Write the gaps
26 − 14 = 12, 38 − 26 = 12, 50 − 38 = 12, 62 − 50 = 12.
- 2
Confirm the rule
All four gaps are 12, so the rule is add 12.
- 3
Take one step
62 + 12 = 74.
- 4
See why the other choices are there
62 repeats the last term. 86 adds 12 twice, which is the term after next. 72 adds 10, a gap that never appears in the series.
Answer
74
Find the wrong step
A student continues this series: 70, 64, 58, 52, …
- 1
Writes the gaps: 64 − 70 = −6, 58 − 64 = −6, 52 − 58 = −6.
- 2
Sees that every gap is the same size, 6.
- 3
Adds 6 to the last term and gets 58.
What went wrong
The gaps are −6, not +6. The series is falling, so the next term must be smaller than 52.
The fix
Subtract 6 from the last term: 52 − 6 = 46.
Step 3 of 5
Find the gap and use it
Check yourself · Question 1
Which number continues the pattern? 33, 41, 49, 57, 65, …
Answer: C
Every gap is +8: 41 − 33, 49 − 41, 57 − 49 and 65 − 57 all equal 8. So the next term is 65 + 8 = 73.
Trap answer: 81
Two steps instead of one
- Why it looks right
- After writing several gaps of 8, it is easy to add the gap a second time without noticing.
- Why it's wrong
- The question asks for the next term, which is one step past 65.
- The right answer
- 73, because 65 + 8 = 73.
Check yourself · Question 2
What number belongs in the blank? 17, 30, ___, 56, 69
Answer: B
The known gaps are 30 − 17 = 13 and 69 − 56 = 13. Also, 56 − 17 = 39 covers three gaps, and 39 ÷ 3 = 13. The blank is 30 + 13 = 43. Check: 43 + 13 = 56.
Common mistake
"The first gap is 8, so the rule is add 8."
- Why it's tempting
- The first gap is quick to find, and on a timed section it feels safe to stop there.
- Do this instead
- A series can start one way and then change. Write every gap and make sure they all match before you choose.
Step 4 of 5
What to remember
Remember
- Subtract each neighbor pair and write the gaps down.
- The rule holds only if every gap matches, including the sign.
- Add the gap to the last term once; the term after next is a common wrong choice.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What were the gaps in the practice series, and which choice would you get by adding the gap twice?