The
Use second differences and subtract the quadratic part.
- Check whether the change is additive, multiplicative or based on the previous two terms.
- A term-to-term rule moves between terms. A position-to-term rule uses the term number
. - Check a proposed rule against at least the first two given terms.
Example 1
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | -2 | 1 | -3 |
| 2 | -3 | 4 | -7 |
| 3 | -2 | 9 | -11 |
| 4 | 1 | 16 | -15 |
| 5 | 6 | 25 | -19 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Example 2
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | -1 | 2 | -3 |
| 2 | 1 | 8 | -7 |
| 3 | 7 | 18 | -11 |
| 4 | 17 | 32 | -15 |
| 5 | 31 | 50 | -19 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Your turn
Question 1Subtract the quadratic part
Find the
Check answer
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 3 | -3 |
| 2 | 5 | 12 | -7 |
| 3 | 16 | 27 | -11 |
| 4 | 33 | 48 | -15 |
| 5 | 56 | 75 | -19 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Question 2Subtract the quadratic part
Find the
Check answer
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | -1 | 1 | -2 |
| 2 | -1 | 4 | -5 |
| 3 | 1 | 9 | -8 |
| 4 | 5 | 16 | -11 |
| 5 | 11 | 25 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Question 3Subtract the quadratic part
Find the
Check answer
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 2 | -2 |
| 2 | 3 | 8 | -5 |
| 3 | 10 | 18 | -8 |
| 4 | 21 | 32 | -11 |
| 5 | 36 | 50 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Question 4Subtract the quadratic part
Find the
Check answer
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 1 | 3 | -2 |
| 2 | 7 | 12 | -5 |
| 3 | 19 | 27 | -8 |
| 4 | 37 | 48 | -11 |
| 5 | 61 | 75 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Question 5Subtract the quadratic part
Find the
Check answer
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 1 | -1 |
| 2 | 1 | 4 | -3 |
| 3 | 4 | 9 | -5 |
| 4 | 9 | 16 | -7 |
| 5 | 16 | 25 | -9 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 3 | -3 |
| 2 | 5 | 12 | -7 |
| 3 | 16 | 27 | -11 |
| 4 | 33 | 48 | -15 |
| 5 | 56 | 75 | -19 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | -1 | 1 | -2 |
| 2 | -1 | 4 | -5 |
| 3 | 1 | 9 | -8 |
| 4 | 5 | 16 | -11 |
| 5 | 11 | 25 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 2 | -2 |
| 2 | 3 | 8 | -5 |
| 3 | 10 | 18 | -8 |
| 4 | 21 | 32 | -11 |
| 5 | 36 | 50 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 1 | 3 | -2 |
| 2 | 7 | 12 | -5 |
| 3 | 19 | 27 | -8 |
| 4 | 37 | 48 | -11 |
| 5 | 61 | 75 | -14 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule
Find the
| Position | Term | Remainder | |
|---|---|---|---|
| 1 | 0 | 1 | -1 |
| 2 | 1 | 4 | -3 |
| 3 | 4 | 9 | -5 |
| 4 | 9 | 16 | -7 |
| 5 | 16 | 25 | -9 |
Black: given data. Blue: working we add.
Second difference
Quadratic part
Remainder rule