Finite Differences

1. Finite Differences The Key To Unlocking The Pattern

2. Patterns may be represented by various rules. Quadratic—Second Differences—an2+bn + c Cubic—Third Differences—an 3+bn2+cn + d Linear—First Differences—an+b Quartic—Fourth Differences—an 4 +bn3+cn2+dn + e This pattern continues but we will not generally study any patterns higher than fourth differences.

3. Given a Pattern0, 16, 64, 162, 328 • STEP 1: Create a table and fill in the values of the given pattern. “n” represents the number of the term (i.e. first term, second term, etc). There will not be a “zero” term at this time so create a spaceholder for that term.

4. Step 1: Create the table n 0 1 0 2 16 3 64 4 162 5 328

5. Given a Pattern0, 16, 64, 162, 328 • STEP 2: Calculate the difference from term 1 to term 2, term 2 to term 3, etc. These are called the first differences. As you work down the table, you add. + As you work up a the table you subtract. -

6. Step 2: Calculate 1st differences n 0 1 0 2 16 3 64 4 162 5 328 Are the first differences equal? 16 48 If yes, then you have a linear pattern. If no, then continue to find the second differences. 98 166

7. Given a Pattern0, 16, 64, 162, 328 • STEP 3: If the first differences are not equal, then create a spaceholder at the top of this column. Then calculate the difference from 16 to 48; from 48 to 98, etc. These are called the second differences. As you work down the table, you add. + As you work up a the table you subtract. -

8. Step 3: Calculate 2nd Differences n 0 1 0 2 16 3 64 4 162 5 328 Are the second differences equal? 16 32 If yes, then you have a quadratic pattern. If no, then continue to find the third differences. 48 50 98 68 166

9. Given a Pattern0, 16, 64, 162, 328 • STEP 4: If the second differences are not equal, then create a spaceholder at the top of this column. Then calculate the difference from 32 to 50 and from 50 to 68. These are called the third differences. As you work down the table, you add. + As you work up a the table you subtract. -

10. Step 4: Calculate 3rd Differences n 0 1 0 2 16 3 64 4 162 5 328 Are the third differencesequal? 16 32 If yes, then you have a cubic pattern. If no, then continue to find the fourth differences. 18 48 50 18 98 68 166

11. Given a Pattern0, 16, 64, 162, 328 • STEP 5: Since the third differences are the same in this case, we know we have a cubic pattern. We now need to work backwards and fill in our shapes that are serving as placeholders. As you work down the table, you add. + As you work up a the table you subtract. -

12. Step 5: Calculate 3rd Differences n 0 1 0 2 16 3 64 4 162 5 328 We know the third differences are all the same (18) so we may fill in this difference to be 18. 18 16 32 18 48 50 18 98 68 166

13. Step 6: Work backwards to fill in the placeholders. n 0 1 0 2 16 3 64 4 162 5 328 We know the third differences are all the same (18) so we may fill in this difference to be 18. 18 16 32 18 48 Now we need to fill in the . Ask yourself 32 – 18 = what? 50 18 98 68 166

14. Step 6: Work backwards to fill in the placeholders. n 0 1 0 2 16 3 64 4 162 5 328 14 We find that the = 14. 18 16 32 Now we need to fill in the . Ask yourself 16 – 14 = what? 18 48 50 18 98 68 166

15. Step 6: Work backwards to fill in the placeholders. n 0 1 0 2 16 3 64 4 162 5 328 2 14 We find that the = 2. 18 16 32 Now we need to fill in the . Ask yourself 0 – 2 = what? 18 48 50 18 98 68 166

16. Step 6: Work backwards to fill in the placeholders. n 0 1 0 2 16 3 64 4 162 5 328 - 2 2 14 We find that the = - 2. 18 16 32 18 48 50 18 98 68 166

17. Given a Pattern0, 16, 64, 162, 328 • PART B Now that we have completed the table, we need to look at the corresponding table that fits a third finite difference pattern. Every third difference can be modeled by the rule: an 3+bn2+cn + d As you work down the table, you add. + As you work up a the table you subtract. -

18. Step 1: Create the table by substituting the value of n. n an 3+bn2+cn + d 0 1 a(1)3+b(1)2+c(1) + d 2 a(2)3+b(2)2+c(2) + d 3 a(3)3+b(3)2+c(3) + d 4 a(4)3+b(4)2+c(4) + d 5 a(5)3+b(5)2+c(5) + d

19. This is the simplified table. Now we will calculate the first differences just like we did in the last table. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d 7a +3b+c 19a +5b+c 37a +7b+c 61a +9b+c

20. This is the simplified table. Now we will calculate the second differences just like we did in the last table. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d 7a +3b+c 12a+2b 19a +5b+c 18a+2b 37a +7b+c 24a+2b 61a +9b+c

21. This is the simplified table. Now we will calculate the third differences just like we did in the last table. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d 7a +3b+c 6a 12a+2b We know the third differences are all the same (6a) so we may fill in this difference to be 6a. 19a +5b+c 6a 18a+2b 6a 37a +7b+c 24a+2b 61a +9b+c

22. Now we will work backwards to fill in the shapes. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d 6a+2b 7a +3b+c 6a 12a+2b 19a +5b+c 6a 18a+2b 6a 37a +7b+c 24a+2b 61a +9b+c

23. Now we will work backwards to fill in the shapes. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d a+b+c 6a+2b 7a +3b+c 6a 12a+2b 19a +5b+c 6a 18a+2b 6a 37a +7b+c 24a+2b 61a +9b+c

24. Now we will work backwards to fill in the shapes. n an 3+bn2+cn + d 0 1 a + b + c + d 2 8a+4b+2c + d 3 27a+9b+3c + d 4 64a+16b+4c+d 5 125a+25b+5c+d a+b+c d 6a+2b 7a +3b+c 6a 12a+2b 19a +5b+c 6a 18a+2b 6a 37a +7b+c 24a+2b 61a +9b+c

25. Now we will compare the values in the shapes for the two tables. Table for the Given Pattern Table for 3rd Differences Conclusion 18 = 6a a= 3 18 6a 6a+2b = 14 6(3)+2b=14 b= -2 14 6a+2b a+b+c 2 a+b+c= 2 3+ -2 +c = 2 1+c=2 c=1 d = -2 d - 2

26. Now that we know the values of a, b, c, and d, we can write the rule. an 3+bn2+cn + d Rule =3n3+ (-2) n2+1n + (-2) Rule = 3n3-2n2+n -2 We can now use this rule to calculate any specific term. For instance, if we want to know the 11th term, we substitute n = 11. Term 11= 3(11)3-2(11)2+ 11 -2 Term 11 = 3760

27. This same procedure will be followed for any difference. For first differences, you stop whenever the first differences are =. You will be finding a and b. For second differences, you stop whenever the second differences are =. You will be finding a, b, and c.