Patterns that multiply
Doubling, halving and multiplying by 3: patterns that grow fast, how to fill a gap in them, and their 10th number.
Some patterns do not add the same number each time. They multiply by the same number each time, and that makes them grow very fast. There is an old story about a king and a chessboard. One grain of rice is put on the first square, 2 on the second, 4 on the third, and the amount doubles on every square. The last of the 64 squares alone would need more than 9 million million million grains, and all the squares together about twice that. That is far more rice than the whole world grows in a year. Patterns like this also go the other way: halving again and again, as when you fold a sheet of paper in half and then in half again.
Adding or multiplying?
First find out what kind of pattern you have. Write the gaps under the numbers. If the gaps are all the same, it is an adding pattern. If the gaps keep getting bigger, try dividing each number by the one before it.
| Pattern | 5 | 10 | 20 | 40 | ? |
|---|---|---|---|---|---|
| Gap | start | +5 | +10 | +20 | ? |
| Times | start | ×2 | ×2 | ×2 | ×2 |
The gaps are 5, 10, 20, so they are not the same. But 10 ÷ 5 = 2, 20 ÷ 10 = 2 and 40 ÷ 20 = 2. Each number is the one before times 2. The number you multiply by every time is called the multiplier. Here the next number is 40 × 2 = 80.
Doubling
Doubling means multiplying by 2, or adding a number to itself. Doubling patterns are the most common multiplying patterns.
Example 1
What number comes next? 13, 26, 52, 104, ?
- 26 ÷ 13 = 2 and 52 ÷ 26 = 2. Each number is double the one before.
- Double the last number: 104 × 2 = 208.
Answer: 208
Halving
Halving means dividing by 2. A halving pattern gets smaller each time, but each step is smaller than the last. In 96, 48, 24, 12 each number is half of the one before, so the next is 12 ÷ 2 = 6.
Example 2
What number comes next? 2000, 1000, 500, 250, ?
- Each number is half of the one before.
- Halve 250 in parts: 250 is 200 and 50.
- Half of 200 is 100, and half of 50 is 25.
- 100 + 25 = 125.
Answer: 125
Times 3, times 4, times 5
The multiplier can be any number. To find it, divide the second number by the first. Then check it with the next pair.
Example 3
What number comes next? 1, 3, 9, 27, ?
- 3 ÷ 1 = 3, 9 ÷ 3 = 3, 27 ÷ 9 = 3. The multiplier is 3.
- 27 × 3 = 81.
Answer: 81
Example 4
What number comes next? 2, 10, 50, 250, ?
- 10 ÷ 2 = 5 and 50 ÷ 10 = 5. The multiplier is 5.
- 250 × 5 = 1250. (Think of it as 250 × 10 = 2500, then half of that.)
Answer: 1250
Fill a missing number
When a number is missing from a multiplying pattern, find the multiplier from two neighbours that are both there. Multiply forward to fill the gap, then check that one more step lands on the next number.
Example 5
What number is missing? 2, 6, ?, 54
- 6 ÷ 2 = 3, so the multiplier is 3.
- 6 × 3 = 18.
- Check: 18 × 3 = 54. It fits.
Answer: 18
If the pattern is going down, the multiplier works backwards: you divide. In 243, 81, ?, 9, 3, each number is the one before divided by 3. So the missing number is 81 ÷ 3 = 27, and 27 ÷ 3 = 9 checks it.
Jump to the 10th number
In an adding pattern, the 10th number is the first number plus 9 steps. In a multiplying pattern it is the first number multiplied by the multiplier 9 times. We write repeated multiplying as a power: 2³ means 2 × 2 × 2 = 8. The small raised number says how many times to multiply.
| Place | 1st | 2nd | 3rd | 4th | 10th |
|---|---|---|---|---|---|
| Number | 2 | 4 | 8 | 16 | ? |
| As a power | 2¹ | 2² | 2³ | 2⁴ | 2¹⁰ |
In 2, 4, 8, 16, ... the number in each place is 2 to the power of the place. So the 10th number is 2¹⁰ = 1024.
Remember
number in place n = first number × (multiplier to the power n − 1)
The power is one less than the place, because the first number has not been multiplied yet. For 2, 4, 8, 16 this gives 2 × 2⁹ = 2¹⁰, the same answer.
Example 6
What is the 10th number in 3, 6, 12, 24, ...?
- The first number is 3 and the multiplier is 2.
- From the 1st place to the 10th, you multiply 9 times: 2⁹ = 512.
- 3 × 512 = 1536.
Answer: 1536
Try it yourself
1.What number comes next? 7, 14, 28, 56, ?
Show the answer
112
Each number is double the one before. 56 × 2 = 112.
2.What number comes next? 320, 160, 80, 40, ?
Show the answer
20
Each number is half of the one before. 40 ÷ 2 = 20.
3.What number comes next? 3, 15, 75, ?
Show the answer
375
The multiplier is 15 ÷ 3 = 5. 75 × 5 = 375.
4.What number is missing? 5, ?, 45, 135
Show the answer
15
135 ÷ 45 = 3, so the multiplier is 3. 5 × 3 = 15, and 15 × 3 = 45 fits.
5.What is the 8th number in 1, 2, 4, 8, ...?
Show the answer
128
The first number is 1 and you double 7 times: 2⁷ = 128.
6.What is the 7th number in 5, 10, 20, 40, ...?
Show the answer
320
You double 6 times after the first number. 2⁶ = 64, and 5 × 64 = 320.
Ready to test yourself?
Play Patterns: 20 seconds per question, and it gets harder as you go.