HSPT: Number series: when the gaps grow
Learn to continue a series whose gaps change by a steady amount, by finding the next gap first.
How do you continue a series when the differences between terms keep changing?
Short answer
Write the gaps, then see how the gaps change. Extend the gaps by one step and add that new gap to the last term.
Know first: Constant-difference series, Subtracting whole numbers
Step 1 of 5
When the gaps change, look at the gaps of the gaps
Sometimes the gaps between terms are not equal, but they change in a steady way: 3, 6, 9, 12, or 1, 3, 5, 7. Then the series has growing differences.
Write the gaps in a row under the series. Then find the gap between those gaps. That second step tells you the next gap, and the next gap gives you the next term.
| Series | Gaps | Gaps grow by | Next term |
|---|---|---|---|
| 40, 41, 43, 46, 50 | 1, 2, 3, 4 | 1 | 50 + 5 = 55 |
| 2, 4, 9, 17, 28 | 2, 5, 8, 11 | 3 | 28 + 14 = 42 |
| 50, 45, 41, 38, 36 | −5, −4, −3, −2 | 1 | 36 − 1 = 35 |
Find the next gap first, then add it to the last term.
Step 2 of 5
Predict the next gap first
Worked example: Gaps that grow by 3
Which number continues the pattern? 12, 15, 21, 30, 42, …
- A54
- B57
- C60
- D45
- 1
Write the gaps
3, 6, 9, 12.
- 2
Find how the gaps change
Each gap is 3 more than the one before.
- 3
Get the next gap
12 + 3 = 15.
- 4
Take one step
42 + 15 = 57.
- 5
See why the other choices are there
54 adds the last gap, 12, again. 60 adds 18, which skips a gap. 45 adds 3, the first gap, as if the series had a constant difference.
Answer
57
Wrong: The last gap was 12, so the next term is 42 + 12 = 54.
Right: The gaps grow by 3, so the next gap is 15 and the next term is 42 + 15 = 57.
In a growing-differences series, the gap itself changes. Reusing the last gap freezes a pattern that is still moving.
Step 3 of 5
Extend the gaps, then the series
Check yourself · Question 1
Which number continues the pattern? 5, 6, 9, 14, 21, …
Answer: A
The gaps are 1, 3, 5, 7, growing by 2. The next gap is 9, so the next term is 21 + 9 = 30.
Trap answer: 28
Frozen gap
- Why it looks right
- 7 is the most recent gap, and adding it looks just like a constant-difference series.
- Why it's wrong
- The gaps 1, 3, 5, 7 are still rising by 2, so the next gap is 9.
- The right answer
- 30, because 21 + 9 = 30.
Check yourself · Question 2
Which number continues the pattern? 3, 8, 18, 33, 53, …
Answer: D
The gaps are 5, 10, 15, 20, growing by 5 each time. The next gap is 25, so the next term is 53 + 25 = 78.
Common mistake
"The gaps are not equal, so this isn't a pattern I know."
- Why it's tempting
- The first test, equal gaps, fails, and it feels like you have run out of ideas.
- Do this instead
- Look at the gaps themselves. If they rise by the same amount each time, extend them by one more step.
Step 4 of 5
What to remember
Remember
- If the gaps are not equal, write the gaps of the gaps.
- Find the next gap before you find the next term.
- Reusing the last gap is the most common wrong choice in this pattern.
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 what would the next gap be?