Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers



Possible Solutions: Supplementary Problems Set F


Problem Set F
Look at possible solutions to Set F problems.

1.

Generate a difference table for the linear function y = 3x - 5 for 1 <= x <= 8.

x

1

2

3

4

5

6

7

8

f(x)

-2

1

4

7

10

13

16
19

D1 --->

3
3
3
3
3
3
3

.

2.

Generate a difference table for the general linear polynomial y = ax + b for 1 <= x <= 8.

x

1

2

3

4

5

6

7

8

f(x)

a+b

2a+b

3a+b

4a+b

5a+b

6a+b

7a+b
8a+b

D1 --->

a
a
a
a
a
a
a

.

3.

Repeat problem 1 for the cubic polynomial y = 2x^3 - x^2 + 3x + 1 for 1 <= x <= 8.

x

1

2

3

4

5

6

7

8

f(x)

5

19

55

125

241

415

659
985

D1 --->

14
36
70
116
174
244
326

.

D2 --->

22
34
46
58
70
82

.

D3 --->

12
12
12
12
12

.

4.

Repeat problem 2 for the general cubic polynomial y = ax^3 + bx^2 + cx + d for 1 <= x <= 8.

x

1

2

3

4

5

6

7

8

f(x)

a+b+c+d

8a+4b+2c+d

27a+9b+3c+d

64a+16b+4c+d

125a+25b+5c+d

216a+36b+6c+d

343a+49b+7c+d
512a+64b+8c+d

D1 --->

7a+3b+c
19a+5b+c
37a+7b+c
61a+9b+c
91a+11b+c
127a+13b+c
169a+15b+c

.

D2 --->

12a+2b
18a+2b
24a+2b
30a+2b
36a+2b
42a+2b

.

D3 --->

6a
6a
6a
6a
6a

.

5.

Complete the difference table for the values in the function y = f(x) shown here.

x

1

2

3

4

5

6

f(x)

12

28

50

78

112

152

D1 --->

16
22
28
34
40

..

D2 --->

6
6
6
6

..

What type of function is this? How do you know?

Based on the information in the table, f(x) is a quadratic function because a constant difference appears in the second-differences row of the differences table.

6.

Determine an explicit representation for the relationship defined in problem 5.

We first create a difference table for the general quadratic function.

x

1

2

3

4

5

6

7

8

f(x)

a+b+c

4a+2b+c

9a+3b+c

16a+4b+c

25a+5b+c

36a+6b+c

49a+7b+c
64a+8b+c

D1 --->

3a+b
5a+b
7a+b
9a+b
11a+b
13a+b
15a+b

D2 --->

2a
2a
2a
2a
2a
2a

D3 --->

.
.
.
.
.

We now equate specific values from the difference table in (5) to the general values in the current difference table. We have that 2a=6 which implies that a=3. Using a=3 in the first-differences row, we get 9+b=16, or b=7. Finally, with b=7 and a=3, we have 10+c=12, or c = 2.

The explicit quadratic representation for the relationship described in (5) is y = f(x) = 3x^2 + 7x + 2. You should calculate some values of f(x) and compare to the values in the second line of the difference table in (5).

7.

The table here shows values for x greater than 2. Use the method of constant differences to determine an explicit representation for the relationship shown and then use it to determine f(1) and f(2), assuming the explicit relationship holds for f(1) and f(2).

x

3

4

5

6

7

8

f(x)

64

141

266

451

708

1049

x

1

2

3

4

5

6

7

8

f(x)

?

?

64

141

266

451

708
1049

D1 --->

?
?
77
125
185
257
341

D2 --->

?
?
48
60
72
84

D3 --->

?
?
12
12
12

We infer that the relationship y=f(x) is cubic, based on the row of constant differrences in line D(3). We use the general cubic difference table (question 4) and solve for a, b, c, and d:

x

1

2

3

4

5

6

7

8

f(x)

a+b+c+d

8a+4b+2c+d

27a+9b+3c+d

64a+16b+4c+d

125a+25b+5c+d

216a+36b+6c+d

343a+49b+7c+d
512a+64b+8c+d

D1 --->

7a+3b+c
19a+5b+c
37a+7b+c
61a+9b+c
91a+11b+c
127a+13b+c
169a+15b+c

D2 --->

12a+2b
18a+2b
24a+2b
30a+2b
36a+2b
42a+2b

D3 --->

6a
6a
6a
6a
6a

We have 6a=12 so that a=2. Then 24a+2b=48, and with a=2, b=0. Also, 37a+7b+c=77, and with a=2 and b=0 we get c=3. Finally, 27a+9b+3c+d=64, ands with a=2, b=0, c=3, we get d=1. This checks for oth values of (x,f(x)).

The desired cubic representation is y = f(x) = 2x^3 + 3x + 1 for the values shown here. This cubic equation yields f(1) = 6 and f(2) = 23.

8.

Determine an explicit representation for the relationship shown in the table below.

x

1

2

3

4

5

6

.

h(x)

4

23

86

247

584

1199

Create a difference table for this sequence:

x

1

2

3

4

5

6

h(x)

4

23

86

247

584

1199

D1 --->

19
63
161
337
615

D2 --->

44
98
176
278

D3 --->

54
78
102

D4 --->

24
24

The constant differences in row D4 implies that a 4th-degree, or quartic, polynomial can model the relationship between x and h(x). We need the general 4th-degree difference table, generated for f(x)=ax^4+bx^3+cx^2+dx+e

x

1

2

3

4

5

f(x)

a+b+c+d+e

16a+8b+4c+2d+e

81a+27b+9c+3d+e

256a+64b+16c+4d+e

625a+125b+25c+5d+e

D1 --->

15a+7b+3c+d
65a+19b+5c+d
175a+37b+7c+d
369a+61b+9c+d

D2 --->

50a+12b+2c
110a+18b+2c
194a+24b+2c

D3 --->

60a+6b
84a+6b

D4 --->

24a

We know equate values between the two tables. This results in a=1, b=-1, c=3, d=2, and e=-1. The explicit formula for the sequence in the original table is h(x)=x^4 - x^3 + 3x^2 + 2x - 1. Use this to confirm the values of h(x) shown in the table.

9.

The standard multiplication table appears to the left below. The pattern to the right is formed by rotating the multiplication table 45 degrees clockwise. Continue the pattern in the right-hand table for at least two more rows. Then determine a function that can be used to calculate the sum of any row in the right-hand table. Use it to determine the sum of the 100th row in that table.

1

2

3

4

5

6

2

4

6

8

10

12

3

6

9

12

15

18

4

8

12

16

20

24

5

10

15

20

25

30

6

12

18

24

30

36

1

2
2

3
4
3

4
6
6
4

5
8
9
8
5

We create a difference table for the sums of the 5 rows tha are shown.

Row #

1

2

3

4

5

Sum

1

4

10

20

35

D1 --->

3
6
10
15

D2 --->

3
4
5

D3 --->

1
1

This is precisely the difference table we created in class when we determined that the explicit formula is, for this context,

Sum(row n) = 1/6n^3+1/2n^2+1/3n = (1/6)(n^3+3n^2+2n) = (1/6)(n)(n+1)(n+2).

If the pattern in the modified multiplication table continues, the sum of row 100 is 171,700.

10.

Determine an explicit representation for the sum of any row in the infinite array whose first three lines are shown here.

(1 x 2) =
(1 x 2) + (2 x 3) =
(1 x 2) + (2 x 3) + (3 x 4) =

Again we create a difference table for the row sums. You should verify the sums for rows 4 and 5 shown here.

Row #

1

2

3

4

5

Sum

2

8

20

40

70

D1 --->

6
12
20
30

D2 --->

6
8
10

D3 --->

2
2

This implies that the pattern results from a cubic relationship. We return to the difference table for the general cubic polynomial and equate table entries.

x

1

2

3

4

5

f(x)

a+b+c+d

8a+4b+2c+d

27a+9b+3c+d

64a+16b+4c+d

125a+25b+5c+d

D1 --->

7a+3b+c
19a+5b+c
37a+7b+c
61a+9b+c

D2 --->

12a+2b
18a+2b
24a+2b

D3 --->

6a
6a

We determine that a=1/3, b=1, c=2/3, and d=0. This yields an explicit formula for the sum of any row in the table, assuming the pattern in the table continues as is shown here. The formula is

Sum(row n) = 1/3n^3 + n^2+2/3n = (1/3)(n^3+3n^2+2n) = (1/3)(n)(n+1)(n+2).

By inspection we see that this formula holds for the values in the table.

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