Arithmetic Progressions
What is an AP
Recognise an arithmetic progression and its common difference.
Patterns are everywhere: the 3-times table (3, 6, 9, 12 …), Reena's salary (₹8000, ₹8500, ₹9000 … rising by ₹500 a year), the rungs of a ladder getting shorter by the same amount.
An arithmetic progression (AP) is a list of numbers where you get each term by adding the same fixed number to the one before. That fixed number is the common difference, d.
| List | d | Kind |
|---|---|---|
| 8000, 8500, 9000 … | +500 | going up |
| 45, 43, 41 … | –2 | going down |
| 3, 3, 3 … | 0 | staying the same |
Not every pattern is an AP. The test is always: is the same number added each time?
- 10000, 12500, 15625 … multiplies by 5/4 each time: not an AP.
- 1, 4, 9, 16 … (squares) adds 3, then 5, then 7: not an AP.
- 1, 1, 2, 3, 5, 8 … (the rabbit pattern) adds the two previous terms: not an AP.
The general form: a, a + d, a + 2d, a + 3d, …, where a is the first term.
Two numbers fix the whole AP: a and d. With a = 4 and d = –3: 4, 1, –2, –5, … Change d and you get a different AP: a = 4, d = 3 gives 4, 7, 10, …
An AP can be finite, with a last term (the heights of a class, 147 cm … 157 cm), or infinite: the dots '…' at the end mean it never stops.
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