#### Patterns, tables and the relationship between numbers.

1. Consider the sequence: 2 ; 5; 8; a; b; . . .
1.1 Each number is 3 more than its predecessor / Each number is equal to the sum of its predecessor and 3.
1.2 a = 11 and b = 14
1.3 Sequence: 2; 5; 8; 11; 14; 17; 20; 23
The 8th number in the sequence = 23
1.4 The 6th number.

2. Consider the number pattern: –4 ; –1; 2; 5; f; g; . . .
2.1 Each number is 3 more than its predecessor.
2.2 f = 8 and g = 11
2.3 Pattern: –4; –1; 2; 5; 8; 11; 14; 17; 20; 23; 26
The 7th number in the pattern = 14
2.4 The 11th number.

3. Consider the sequence: 4 ; 2; 0; a; b; . . .
3.1 Each number is 2 smaller than its predecessor. / Each number is equal to its predecessor minus 2.
3.2 a = –2 ; b = –4
3.3 Sequence: 4; 2; 0; –2; –4; –6; –8; –10; –12; –14; –16; . . .
The 11th number in the sequence is –16
3.4 The 9th number is equal to –12.

4. Consider the sequence: –8 ; –13; –18; p; q; . . .
4.1 Each number is 5 less than its predecessor. / Each number is equal to its predecessor minus 5.
4.2 p = –23 and q = –28
4.3 Sequence: –8 ; –13; –18; –23; –28; –33; –38; –43; –48; –53; –58; –63; . . .
The 8th number in the sequence is –43
4.4 The 12th number is –63

5. Consider the number pattern: 118; 124; 130; r; s; . . .
5.1 Each number is 6 more than its predecessor. / Each number is equal to its predecessor plus 6.
5.2 r = 136 and s = 142
5.3 Pattern: 118; 124; 130; 136; 142; 148; 154; 160; 166; 172; 178; ...
The 9th number in the pattern is 166.
5.4 The 11th number is 178.
5.5 The 11th number is 178, 12th number = 184; 13th number = 190; 14th = 196; 15th = 202.
The first number greater than 200 is 202 and it is number 15.

6. Consider the sequence: 2; 4; 8; 16; a; b; . . .
6.1 The nect number in the sequence is formed by multiplying its predecessor by 2    OR
the product of its predecessor and 2.
6.2 a = 16 x 2 = 32 and b = 32 x 2 = 64.
6.3 Sequence : 2; 4; 8; 16; 32; 64; 128; 256; . . . the 8th number = 256
6.4 Sequence : 2; 4; 8; 16; 32; 64; 128; 256; 512; 1024; 2048; . . . the 11th number = 2048

7. Consider the pattern: 3; 9; 27; c; d; . . .
7.1 The next number is the product of its predecessor and 3
OR The next number is formed by multiplying its predecessor by 3.
7.2 c = 27 x 3 = 81 and d = 81 x 3 = 243.
7.3 pattern: 3; 9; 27; 81; 243; 729; 2 187; . . . the 7th number = 2 187
7.4 pattern: 3; 9; 27; 81; 243; 729; 2 187; 6 561; 19 683 . . . The 9th number = 19 683

8. Consider the sequence: 768; 384; 192; f; g; . . .
8.1 The next number in the sequence is formed by dividing its predecessor by 2
OR by the quotient of its predecessor and 2.
8.2 f = 192 ÷ 2 = 96 and g = 96 ÷ 2 = 48.
8.3 Sequence: 768; 384; 192; 96; 48; 24; 12; . . . The 7th number = 12
8.4 Sequence: 768; 384; 192; 96; 48; 24; 12; 6; 3; 1,5 ; 0,75 ; . . .
The first number smaller than 1 is 0,75 and it is number 11
8.5 Sequence: 768; 384; 192; 96; 48; 24; 12; 6; 3; 1,5 ; 0,75; 0,375; 0,1875 Number in position 13 = 0,1875

9. Consider the series: 1; -2; 4; -8; p; q; . . .
9.1 Each number is equal to its predecessor multiplied by -2. / Each number is equal to
the product of its predecessor and -2.
9.2 p = 16 and q = -32
9.3 Sequence: 1; -2; 4; -8; 16; -32; 64; -128; 256; -512; 1024; -2048; . . .
The 10th number in the sequence is -512
9.4 The 12th number is -2048.

10. Consider the sequence: -4; -1; 2; 5; s; t; . . .
10.1 Each number is 3 more than its predecessor. / Each number is equal to the sum
of its predecessor and 3.
10.2 s = 8 and t = 11
10.3 Sequence: -4; -1; 2; 5; 8; 11; 14; 17; 20; 23; 26; 29; . . .
The 10th number in the sequence is 23.
10.4 The 9th number is 20.

11. Consider the sequence: 576; 288; 144; v; w; . . .
11.1 Each number is equal to half of its predecessor. / Each number is equal to its
predecessor divided by 2.
11.2 v = 72 and w = 36
11.3 Sequence: 576; 288; 144; 72; 36; 18; 9; 4,5; 2,25; 1,125; 0,5625; 0,28125; 0,140625; . . .
The 10th number in the sequence is 1,125
11.4 The 8th number is 4,5.
11.5 The 11th number is the first number smaller than 1.
11.6 No number can be smaller than 0 because a positive number divided by +2 can not
be smaller than 0, i.e. negative.

12. Consider the sequence: 0,15; 0,6; 2,4; f; g; . . .
12.1 Each number is equal to 4 times its predecessor. / Each number is equal to its
predecessor multiplied by 4.
12.2 f = 9,6 and g = 38,4
12.3 Sequence: 0,15; 0,6; 2,4; 9,6; 38,4; 153,6; 614,4; . . .
The 6th number in the series is 153,6.
12.4 The 7th number. (614,4) is the first number that is greater than 300.

13.1 Complete the following table :

 Position number 1 2 3 4 5 8 12 Number 4 7 10 13 16 25 37
13.2 A number is formed by adding 3 to its predecessor. Keep on adding 3 until the required number is reached.
a = 10 + 3 = 13 and b = 13 + 3 = 16 .
13.3 Expand the pattern by adding 3 until the required number is reached. Then find the position number.
Pattern : 4; 7; 10; 13; 16; 19; 22; 25; 28; 31; 34; 37     c = 8 and d = 12

14.1 Complete the following table :

 Position number 1 2 3 4 5 c d 9 e Getal 58 51 44 f g 23 16 h − 5
14.2 The next number in the pattern is formed by subtracting 7 from its predecessor    OR
by adding −7 to its predecessor.
Repeatedly add –7 until the required number is reached. Now find the position of the number.
Pattern : 58; 51; 44; 37; 30; 23; 16; 9; 2; -5; -12; . . .    c = 6 ; d = 7 and e = 10.
14.3 Expand the sequence as explained in 14.2 to find the required numbers.
Pattern : 58; 51; 44; 37; 30; 23; 16; 9; 2; -5; -12;     f = 37; g = 30 and h = 2.

15.1 Complete the following table :

 Position number 1 2 3 4 j 6 k Getal 4 12 36 m 324 n 2916
15.2 Multiply the predecessor by 3    OR    The next number is the product of its predecessor and 3.
Sequence : 4; 12; 36; 108; 324; 972; 2 916; 8 748; 26 244; . . .     j = 5 and k = 7.
15.3 Expand the sequence as explained in 15.2 until the required numbers are reached and write them down.
Sequence : 4; 12; 36; 108; 324; 972; 2 916; 8 748; 26 244; . . .     m = 108 and n = 972.

16.1 Complete the following table :

 Position number 1 2 3 4 p 7 q Getal 2 8 32 s 512 t 131072
16.2 The next number is equal to the product of its predecessor and 4. Expand until the required numbers are reached.
Pattern : 2; 8; 32; 128; 512; 2048; 8192; 32 768; 131 072      p = 5 and q = 9.
16.3 Expand the pattern until the required numbers are reached.
Pattern : 2; 8; 32; 128; 512; 2048; 8192; 32 768; 131 072      s = 128 and t = 8 192 .

17.1 Complete the following table :

 Position number 1 2 3 4 v 8 w Getal 128 64 32 x 4 y 0,125
17.2 The next number is equal to the product of its predecessor and 0,5 or divide the predecessor by 2.
Expand until the required numbers are reached. Pattern : 128; 64; 32; 16; 8; 4; 2; 1; 0,5; 0,25; 0,125; 0,0625 v = 6 and w = 11 .
Pattern : 128; 64; 32; 16; 8; 4; 2; 1; 0,5; 0,25; 0,125; 0,0625     v = 6 and w = 11.
17.3 x = 16 en y = 1.
17.4 The number in position 13 is smaller than 0,05. Its value is 0,03125.

18. The numbers in the pattern are formed by adding 3 to the predecessor.
18.1 The first number is 6. Complete the table.

 Position number 1 2 3 4 5 6 7 Number 6 9 12 15 18 21 24 Number of 3's added to 6 0 1 2 3 4 5 6
18.3 Each is 1 less than the number in the first row.
18.4 We call the first number in the pattern a. In this case a = 6
We call the number giving the position in the pattern n. For a   n = 1 (first number in the pattern)
18.5 The second number can be written as 6 + 1 x 3 or as 6 + (2 - 1) x 3
18.6 The third number can be written as 6 + 2 x 3 or as 6 + (3 - 1) x 3
18.7 The sixth number can be written as 6 + 5 x 2 or as 6 + (6 - 1) x 3)
18.8 The nth number can be written as 6 + (n - 1) x 3 and this gives us a way / a formula to calculate the value of
a number in the pattern.
18.9 The 7th number = 6 + (7 - 1) x 3 = 6 + 18 = 24. Yes, the values correspond.
18.10 The 12th number = 6 + (12 - 1) x 3 = 6 + 33 = 39
18.11 The 16th number = 6 + (16 - 1) x 3 = 6 + 45 = 51 which is greater than 48.

19. The numbers in a pattern are formed by adding 7 continously to the predecessor.
19.1 The first number is 8. Complete the table.

 Position number 1 2 3 4 5 6 Number 8 15 22 29 36 43 Number of 7's added to 8 0 1 2 3 4 5
19.3 a = 8
19.4 The number that we add every time is called the common difference and is represented by a d. In this case d = 7
19.5 The third number can be written as 8 + 2 x 7 or as 8 + (3 - 1) x 7
or also as a + (n - 1)d where a = 8, n = 3 (the third number) and d = 7 (7 is added every time).
19.6 Thus the formula is : The nth number = 8 + (n - 1) x 7
The 9th number = 8 + (9 - 1) x 7 = 8 + 56 = 64
19.7 The 22nd number = 8 + (22 - 1) x 7 = 8 + 140 = 148
19.8 The nth number = 8 + (n - 1) x 7 and thus 8 + (n - 1) x 7 = 78
8 + 7n − 7 = 78       . . . remove brackets - multiply
7n + 1 = 78       . . . simplify
7n = 77       . . . subtract 1 both sides
n = 11            The 11th number is equal to 78.
OR The 9th number = 64 (calculated in 19.6) The 10th number is thus = 64 + 7 = 71 and
the 11th number = 71 + 7 = 78 and thus the 11th number = 78

20. The first number in a sequence is 65. A number in the sequence is formed by subtracting 6 from its predecessor.
20.1 Complete the table.

 Position number 1 2 3 4 5 6 Number 65 59 53 47 41 35
20.2 a = 65 ; d = −6
20.3 nth number = 65 + (n - 1) x (−6)
20.4 5th number = 65 + (5 - 1) x (−6) = 65 − 24 = 41 and the 6th number = 65 + (6 - 1) x (−6) = 65 − 30 = 35
20.5 The 9th number = 65 + (9 - 1) x (−6) = 65 − 48 = 17
20.6 nth number = −1 thus 65 + (n - 1) x (−6) = −1
65 + n x −6 + (−1) x −6 = −1
65 −6n + 6 = −1
−6n = −72
n = 12     the 12th number is equal to −1

21. The first number in a sequence is 5. A number in the sequence is formed by multiplying its predecessor by 2.
21.1 Complete the table.

 Position number 1 2 3 4 5 6 Number 5 10 20 40 80 160
21.2 a = 5
21.3 The number by which the predecessor is multiplied is called the common ratio and it is represented by a   r.

22. The first number in a sequence is 1 024. A number in the sequence is formed by dividing its predecessor by 4.
22.1 Complete the table.

 Position number 1 2 3 4 5 Number 1 024 256 64 16 4
22.2 a = 1024 and r = 0,25
22.3 The 7th number is the first number smaller than 1 and its value is 0,25.

23. Given the formula: b = 3a - 5.
23.1 b is equal to the difference between 3 times a and 5. / b is equal to the product of
3 and a subtract 5.
23.2 a is the independent variable and b is the dependent variable.
23.3
 a -6 -2 0 2 8 b -23 -11 -5 1 19

 23.4   b = 3(-16) - 5 23.5 4 = 3a - 5 = -48 - 5 4 + 5 = 3a = -53 3 = a

24. Given the formula: c = 2 - 3d
24.1 c is equal to the difference between 2 and 3 times d. / c is equal to 2 minus the product of
3 and d.
24.2 d is the independent variable and c is the dependent variable.
24.3
 d -5 -3 0 2 6 c 17 11 2 -4 -16

 24.4   c = 2 - 3(-6) 24.5 5 = 2 - 3d = 2 + 18 3d = 2 - 5 = 20 d = -1

25. Given the formula: p = 6 - 7q
25.1 p is equal to the difference between 6 and 7 times q. / p is equal to 6 minus the product of
7 and q.
25.2 q is the independent variable and p is the dependent variable.
25.3
 q -5 -2 0 3 5 p 41 20 6 -15 -29

 25.4   p = 6 - 7(-2) 25.5 -22 = 6 - 7q = 6 + 14 7q = 6 + 22 = 20 q = 4
26. Given the formula: y = 2x - 7
26.1 y is equal to twice x decreased by 7 / y is equal to 2 times x minus 7.
26.2 x is the independent variable and y is the dependent variable.
26.3 Calculate at least 4 values of x and of y and write down in a table.

 x −5 −2 −1 0 4 6 y −17 −11 −9 −7 1 5
26.4 No, it is an increasing / a rising relationship. If x increases, y also increases in the same proportion.

 26.5 y = 2(−3) - 7 26.6 3 = 2x - 7 y = −6 - 7 3 + 7 = 2x y = −13 5 = x

27. Given the formula : xy = 36
27.1 The product of x and y is equal to 36   OR   y is equal to 36 divided by x   OR   x is equal to 36 divided by y
27.2 x is the independent variable and y is the dependent variable.
27.3 Calculate at least 4 values of x and of y and write down in a table.

 x −1 −3 −6 2 4 12 y −36 −12 −2 18 9 3
27.4 No. This is an example of an indirect or inverse proportion because the product of x and of y is a constant.
27.5 The value of y decreases if the value of x increases / becomes greater in the same proportion because
the product of x and y is constant.
27.6 No. xy = 36 and if x = 0 then xy MUST be equal to 0 and if y = 0 the product xy MUST be equal to 0.
OR also x = 36 divided by y and the divisor may not be 0.

 27.7   xy = 36  :  (3)y = 36 27.8 xy = 36  :  x(10) = 36 y = 12 x = 3,6

28. Given the formula: pq = 24
28.1 The product of p and q is equal to 24   OR   q is equal to 24 divided by p   OR   p is equal to 24 divided by q
28.2 p is the independent variable and q is the dependent variable.
28.3 Calculate at least 4 values of p and of q and write down in a table.

 p −8 −3 −1 2 4 12 q −3 −8 −24 12 6 2
28.4 No. This is an example of an indirect / inverse proportion because the product of p and q is a constant.
28.5 The value of q decreases if the value of p increases / becomes greater in the same proportion because
the product of p and q is a constant.
28.6 No. pq = 24 and if p = 0 then pq MUST be equal to 0 and if q = 0 then pq MUST be equal to 0
OR also p = 24 divided by q and the divisor may not be 0.

 28.7   pq = 24  :  (12)q = 36 28.8 pq = 24  :  p(6) = 24 q = 2 p = 4

29. Given the formula: ab = 18
29.1 The product of a and b is equal to 18   OR   b is equal to 18 divided by a   OR   a is equal to 18 divided by b
29.2 a is the independent variable and b is the dependent variable.
29.3 Calculate at least 4 values of a and of b and write down in a table.

 a −18 −5 −3 2 6 10 b −1 −3,6 −6 9 3 1,8
29.4 No. This is an example of an indirect / inverse proportion because the product of p and q is a constant.
29.5 The value of b increases if the value of a decreases / becomes smaller in the same proportion because
the product of a and b is a constant.
29.6 No. ab = 18 and if a = 0 then ab MUST be equal to 0 and if b = 0 then ab MUST be 0
OR also a = 18 divided by b and the divisor may not be 0.

 29.7   ab = 18  :  (12)b = 18 29.8 ab = 18  :  a(4) = 18 29.9 ab = 18  :  a x a = 18 b = 1,5 a = 4,5 a = √18

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