Equal steps in, equal steps out
Look at 3, 5, 7, 9. Is the pattern linear?
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Step 1 of 4
- Bus times: 0, 7, 14, 21, 28 minutes.
- Each gap is +7. The gaps are equal, so it is linear.
- Paper pieces: 2, 4, 8, 16.
- The gaps are +2, +4, +8. They are not equal, so it is not linear.
The rule
Yes. Each number is 2 more than the one before.
The numbers in a table are its readings. Let the inputs go up by one each time. Find the gap between each reading and the next. A pattern is linear when every gap is the same.
A gap is a reading minus the reading before it. Equal steps in give equal steps out.
Take y = ax + b. Let x go up by 1, from x to x + 1.
- Output at
x:ax + b. - Output at
x + 1:a(x + 1) + b = ax + a + b. - Gap:
(ax + a + b) − (ax + b) = a.
So every gap is a.
| Readings | Gaps | Linear? |
|---|---|---|
| 0, 7, 14, 21, 28 | 7, 7, 7, 7 | yes |
| 2, 5, 8, 11 | 3, 3, 3 | yes |
| 20, 16, 12, 8, 4 | −4 each time | yes |
| 2, 4, 8, 16 | 2, 4, 8 | no |
| 3, 6, 9, 20 | 3, 3, 11 | no |
A falling table can be linear. The gap must stay the same, and a gap can be negative.