Quadratic Sequences

Identifying Quadratic Sequences

First differences: not constant. Second differences: constant.

$1, 4, 9, 16, 25$ → Differences: $3, 5, 7, 9$ → Second differences: $2, 2, 2$ ✓

Finding the Nth Term

Step 1: Second difference = $2a$ → find $a$. Step 2: Subtract $an^2$ from each term. Step 3: The remaining sequence is linear — find its nth term ($bn + c$).

Example

Sequence: $3, 9, 19, 33, 51$

First differences: $6, 10, 14, 18$ Second differences: $4, 4, 4$ → $2a = 4$ → $a = 2$.

Subtract $2n^2$: $3-2, 9-8, 19-18, 33-32, 51-50 = 1, 1, 1, 1, 1$

Linear part: $0n + 1$.

Nth term: $2n^2 + 1$.

Another Example

Sequence: $2, 7, 16, 29, 46$

First diff: $5, 9, 13, 17$. Second diff: $4, 4, 4$ → $a = 2$.

Subtract $2n^2$: $0, -1, -2, -3, -4$. Linear: $-n + 1$.

Nth term: $2n^2 - n + 1$. Check: $n=1$: $2-1+1=2$ ✓