When to use: When the differences of consecutive terms form an AP or GP.

Technique:

1. Compute first differences: $d_k$ = $a_{k+1}$ - $a_k$
2. If $d_k$ is constant: original sequence is AP
3. If $d_k$ forms an AP: second differences are constant, so $a_n$ is quadratic in n
4. If $d_k$ forms a GP: sum the GP to find $a_n$

Example: Sequence 2, 5, 10, 17, 26, ... Differences: 3, 5, 7, 9, ... (AP with d=2) Second differences: 2, 2, 2, ... (constant) So $a_n$ = $An^2$ + Bn + C. Using $a_1$=2, $a_2$=5, $a_3$=10: A+B+C=2, 4A+2B+C=5, 9A+3B+C=10. Solving: A=1, B=0, C=1. So $a_n$ = $n^2$ + 1.