HSPT: Number series: find the multiplier
Learn to recognize a series that multiplies or divides by the same number each step, and continue it.
What do you do when the gaps in a number series keep getting bigger?
Short answer
Divide each term by the one before it. If the result is the same every time, multiply the last term by that number.
Know first: Multiplication facts, Dividing whole numbers
Step 1 of 5
When the gaps grow fast, divide
Some series grow faster and faster. The gaps between terms keep getting bigger, so no single number is added each time. Try dividing each term by the one before it. If you get the same answer every time, the rule is multiply by that number.
The same test works for a series that shrinks by halves or thirds. Then the rule is to divide by the same number each time.
| Test | 5, 10, 20, 40 | Result |
|---|---|---|
| Subtract neighbors | 5, 10, 20 | The gaps change, so it is not adding |
| Divide neighbors | 2, 2, 2 | Same every time: multiply by 2 |
Try subtracting first. If the gaps keep growing, try dividing.
Step 2 of 5
Test the ratio on every pair
Worked example: A series that grows by fours
Which number continues the pattern? 3, 12, 48, 192, …
- A768
- B336
- C196
- D576
- 1
Try the gaps
12 − 3 = 9, 48 − 12 = 36, 192 − 48 = 144. They are not equal.
- 2
Try dividing
12 ÷ 3 = 4, 48 ÷ 12 = 4, 192 ÷ 48 = 4. The rule is multiply by 4.
- 3
Take one step
192 × 4 = 768.
- 4
See why the other choices are there
336 adds the last gap again (192 + 144). 196 adds 4 when the rule says multiply by 4. 576 multiplies by 3.
Answer
768
Find the wrong step
A student continues this series: 5, 10, 20, 40, …
- 1
Subtracts the first pair: 10 − 5 = 5.
- 2
Decides the rule is add 5, so the next term is 40 + 5 = 45.
What went wrong
One gap is not a rule. The next gap is 20 − 10 = 10, which already breaks add 5.
- 3
Chooses 45.
The fix
Each term is double the one before: 10 ÷ 5 = 2, 20 ÷ 10 = 2, 40 ÷ 20 = 2. The next term is 40 × 2 = 80.
Step 3 of 5
Multiply or divide by the same number
Check yourself · Question 1
Which number continues the pattern? 2, 10, 50, 250, …
Answer: B
Each term is 5 times the one before: 10 ÷ 2, 50 ÷ 10 and 250 ÷ 50 all equal 5. The next term is 250 × 5 = 1,250.
Trap answer: 450
Adding the last gap
- Why it looks right
- Adding the most recent gap works for adding patterns, so it feels like a safe habit.
- Why it's wrong
- The gaps here are 8, 40 and 200, so they are not constant. The rule only shows up when you divide.
- The right answer
- 1,250, because 250 × 5 = 1,250.
Check yourself · Question 2
Which number continues the pattern? 896, 448, 224, 112, …
Answer: B
Each term is half the one before: 448 ÷ 896 = 1/2, and so on. The next term is 112 ÷ 2 = 56.
Common mistake
"I subtract the first two terms and keep adding that number."
- Why it's tempting
- Many series questions do use adding, so it becomes a habit.
- Do this instead
- When the gaps keep growing or shrinking, stop and divide neighbors. A steady ratio means multiply.
Step 4 of 5
What to remember
Remember
- Subtract first; if the gaps grow or shrink, divide neighbors instead.
- The rule is real only if the same ratio works for every pair.
- A shrinking series can follow a ratio too: divide by the same number each step.
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
How did you check that the practice series multiplies, and which wrong choice comes from adding instead?