Arithmetic Progressions
Finding a and d
Test whether a list is an AP and find its first term and common difference.
a is simply the first term. To find d, take any term minus the one before it:
d = a₂ – a₁ = a₃ – a₂ = …
Always do later minus earlier, even when the later term is smaller. For 6, 3, 0, –3: d = 3 – 6 = –3, not +3. The list is going down, so d must be negative.
Is this list an AP? Work out every difference, not just one. If they are all equal, it is an AP:
| List | Differences | AP? |
|---|---|---|
| 1, –1, –3, –5 | –2, –2, –2 | yes, d = –2, next: –7, –9 |
| –2, 2, –2, 2 | 4, –4, 4 | no |
| 3, 3 + √2, 3 + 2√2 … | √2, √2 | yes, d = √2 |
Why check every difference? A list can start like an AP and then break: 1, 3, 5, 8 has differences 2, 2, 3.
Once you know a and d, keep adding d to write more terms. With fractions or roots, add them carefully: –1, –½, 0, ½, … (d = ½).
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