HSPT: Number series: split an alternating pattern
Learn to recognize two patterns sharing one series, solve each one, and continue the right one.
What do you do when a number series jumps up and down with no steady gap?
Short answer
Split the terms into odd and even positions and find a rule for each. Then count the terms to see which series the next number continues.
Know first: Constant-difference series, Odd and even numbers
Step 1 of 5
Two series can share one line
In an alternating series, two separate patterns take turns. The terms in the 1st, 3rd and 5th places follow one rule, and the terms in the 2nd, 4th and 6th places follow another.
You'll notice it when the numbers jump up and down with no steady gap. Split the terms by position, solve each pattern, then decide which pattern the next term belongs to.
| Positions | Terms | Rule |
|---|---|---|
| Odd (1st, 3rd, 5th) | 2, 6, 10 | Add 4 |
| Even (2nd, 4th, 6th) | 50, 45, 40 | Subtract 5 |
Six terms are shown, so the next one is the 7th, an odd position: 10 + 4 = 14.
Step 2 of 5
Decide which series comes next
Worked example: Two patterns taking turns
Which number continues the pattern? 11, 60, 14, 52, 17, 44, …
- A20
- B36
- C41
- D17
- 1
Split the terms
Odd positions: 11, 14, 17. Even positions: 60, 52, 44.
- 2
Find each rule
The odd series adds 3. The even series subtracts 8.
- 3
Pick the right series
Six terms are given, so the next is the 7th, an odd position. Continue 11, 14, 17.
- 4
Take one step
17 + 3 = 20.
- 5
See why the other choices are there
36 is the next even term, which comes one place later. 41 uses the odd series' step of 3 on the even series: 44 − 3. 17 repeats the last odd term.
Answer
20
Find the wrong step
A student continues this series: 1, 46, 4, 41, 7, 36, …
- 1
Splits the terms: odd positions 1, 4, 7 and even positions 46, 41, 36.
- 2
Finds the rules: the odd series adds 3 and the even series subtracts 5.
- 3
Says the last term is 36, so the next term continues that series: 36 − 5 = 31.
What went wrong
36 is in the 6th place. The next term is the 7th, which belongs to the odd series.
The fix
Continue the odd series: 7 + 3 = 10.
Step 3 of 5
Split, solve, then choose the series
Check yourself · Question 1
Which number continues the pattern? 8, 70, 13, 63, 18, 56, …
Answer: B
Odd positions: 8, 13, 18, adding 5. Even positions: 70, 63, 56, subtracting 7. The next term is the 7th, an odd position, so it is 18 + 5 = 23.
Trap answer: 49
Wrong series
- Why it looks right
- The last number you read is 56, and its series has a clear rule, so continuing it feels natural.
- Why it's wrong
- Six terms are given, so the next is the 7th. Odd positions belong to the 8, 13, 18 series.
- The right answer
- 23, because 18 + 5 = 23.
Check yourself · Question 2
Which number continues the pattern? 29, 2, 25, 4, 21, 8, 17, …
Answer: C
Odd positions: 29, 25, 21, 17, subtracting 4. Even positions: 2, 4, 8, doubling. Seven terms are given, so the next is the 8th, an even position: 8 × 2 = 16.
Common mistake
"The numbers jump around, so I look at the gap between each neighbor."
- Why it's tempting
- Neighbor gaps are the first thing to try in most series.
- Do this instead
- When the gaps swing between large and small, or between up and down, split the terms by position and look at each half alone.
Step 4 of 5
What to remember
Remember
- Numbers that jump up and down often mean two series taking turns.
- Split by odd and even positions, and find a rule for each.
- Count the terms shown to decide which series the next term continues.
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
In the practice question, which positions held each series, and how did you decide which one comes next?