| 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 |
|
| |
|
| |
|
| |
|
| |
|
| |
|
| |
|
| |
|