| Formula | Expression |
|---|---|
| nth term | a_n = a + (n-1)d |
| Sum of n terms | S_n = n/2 * [2a + (n-1)d] |
| Sum (alternate) | S_n = n/2 * (a + l) where l = last term |
| Number of terms | n = (l - a)/d + 1 |
| Common difference | d = a_{n+1} - a_n |
| Middle term (odd n) | a_{(n+1)/2} = S_n / n |
| AP condition | 2b = a + c |
Useful tricks:
- If S_n = An^2 + Bn + C, then AP only if C = 0. Then a_1 = A+B, d = 2A.
- a_n = S_n - S_{n-1} for n >= 2
- Sum of AP = number of terms * middle term (or average of first and last)
- For 3 terms in AP, assume a-d, a, a+d (sum = 3a automatically)
- For 4 terms in AP, assume a-3d, a-d, a+d, a+3d (common diff = 2d)