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

1. Consider the sequence: 2 ; 5; 8; a; b; . . .
1.1 Explain how the next number in the sequence is formed, e.g. how is 5 formed if we start with 2?
Is your rule also valid for 8?
1.2 Find the values of a and b.
1.3 Write down the 8th number in the sequence.
1.4 Which number in the sequence will be equal to 17?

2. Consider the number pattern: –4 ; –1; 2; 5; f; g; . . .
2.1 Explain how the next number in the pattern is formed.
2.2 Find the values of f en g.
2.3 Write down the 7th number in the pattern.
2.4 Which number in the pattern will be equal to 26?

3. Consider the sequence: 4; 2; 0; a; b; . . .
3.1 Explain how the next number in the sequence is formed.
3.2 Find the values of a en b.
3.3 Write down the 11th number in the sequence.
3.4 Which number in the sequence will be equal to -12?

4. Consider the sequence: –8 ; –13; –18; p; q; . . .
4.1 Explain how the next number in the sequence is formed.
4.2 Find the values of p en q.
4.3 Write down the 8th number in the sequence.
4.4 Which number in the sequence will be equal to -63?

5. Consider the number pattern: 118 ; 124; 130; r; s; . . .
5.1 Explain how the next number in the pattern is formed
5.2 Find the values of r en s.
5.3 Write down the 9th number in the pattern.
5.4 Which number in the pattern will be equal to 178?
5.5 Which number in the pattern will be the first number that is greater than 200 and what is
is this number?

6. Consider the sequence: 2; 4; 8; 16; a; b; . . .
6.1 Explain how the next number in the sequence is formed.
6.2 Determine the value of a and of b.
6.3 Write down the 8th number in the sequence.
6.4 What is the number of the number in the pattern that is equal to 2048?

7. Consider the pattern: 3; 9; 27; c; d; . . .
7.1 Explain how the next number in the pattern is formed.
7.2 Find the value of c and of d.
7.3 Write down the 7th number in the pattern.
7.4 Which number in the pattern is equal to 19 683?

8. Consider the sequence: 768; 384; 192; f; g; . . .
8.1 Explain how the next number in the sequence is formed.
8.2 Determine the value of f and of g.
8.3 Write down the value of the 7th number in the sequence.
8.4 Which number in the sequence will be the first number smaller than 1?
8.5 Write doen the number of the number in the sequence that is equal to 0,1875.

9. Consider the sequence: 1; -2; 4; -8; p; q; . . .
9.1 Explain how the next number in the sequence is formed.
9.2 Find the values of p en q.
9.3 Write down the 10th number in the sequence.
9.4 Which number in the sequence will be equal to -2048?

10. Consider the sequence: –4; –1; 2; 5; s; t; . . .
10.1 Explain how the next number in the sequence is formed.
10.2 Find the values of s en t.
10.3 Write down the 10th number in the sequence.
10.4 Which number in the sequence will be equal to 20?

11. Consider the sequence: 576; 288; 144; v; w; . . .
11.1 Explain how the next number in the sequence is formed.
11.2 Find the values of v en w.
11.3 Write down the 10th number in the sequence.
11.4 Which number in the sequence will be equal to 4,5?
11.5 Which number in the sequence will be the first number that is smaller than 1?
11.6 Which number in the sequence will be the first number that is smaller than 0? Explain your answer.

12. Consider the sequence: 0,15; 0,6; 2,4; f; g; . . .
12.1 Explain how the next number in the sequence is formed.
12.2 Find the values of f en g.
12.3 Write down the 6th number in the sequence.
12.4 Which number in the sequence will be the first number that is greater than 300? What is
this number?

13.1 Complete the following table :

 Position number 1 2 3 4 5 c d Number 4 7 10 a b 25 37
13.2 Explain how you will find the values of a and of b. Write down these values.
13.3 Explain how you will find the values of c and of d. Write down these values.

14.1 Complete the following table :

 Position number 1 2 3 4 5 c d 9 e Number 58 51 44 f g 23 16 h − 5
14.2 Explain how you will find the values of c, d and of e. Write down these values.
14.3 Explain how you will find the values of f, g and of h. Write down these values.

15.1 Complete the following table :

 Position number 1 2 3 4 j 6 k Number 4 12 36 m 324 n 2916
15.2 Explain how you will find the values of c, d and of e. Write down these values.
15.3 Explain how you will find the values of c, d and of e. Write down these values.

16.1 Complete the following table :

 Position number 1 2 3 4 p 7 q Number 2 8 32 s 512 t 131072
16.2 Explain how you will find the values of c, d and of e. Write down these values.
16.3 Explain how you will find the values of c, d and of e. Write down these values.

17.1 Complete the following table :

 Position number 1 2 3 4 v 8 w Number 128 64 32 x 4 y 0,125
17.2 Explain how you will find the values of c, d and of e. Write down these values.
17.3 Explain how you will find the values of c, d and of e. Write down these values.
17.4 What is the number of the first number in the pattern that will be smaller than 0,05?

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

 Number of number in pattern, n 1 2 3 4 5 6 7 Number 6 Number of 3's added to 6 0 1 2
18.3 How do the numbers in the third row compare to that in the first row?
18.4 We call the first number in the pattern a. In this case a = ?
We call the number of the number in the pattern n. For a n = 1.
18.5 We can write the second number as 6 + 1 x 3 or as 6 + (2 - 1) x 3
18.6 We can write the third number as 6 + 2 x 3 or as 6 + (3 - 1) x 3
18.7 We can write the sixth number as ? or as ?
18.8 We can write the nth number as 6 + (n - 1) x 3 and that gives us a method/ formula with which
we can calculate any number in the pattern.
18.9 Now use this formula to calculate the value of the 7th number in the pattern. Does your answer correspond to
the value in the table?
18.10 Use this formula to show that the 12th number in the pattern is equal to 39.
18.11 Show that the 16th number is greater than 48

19. The numbers in a pattern are formed by adding 7 to the previous number.
19.1 The first number in the pattern is 8. Now complete the table.

 n 1 2 3 4 5 6 Number 8 Number of 7's added to 8 0 1 2
19.3 The first number in te pattern is a. In this case a = ?
19.4 The number that we add to form the next number is called the common difference and is represented by a d.
In this case d = ?
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 (we add 7 to get the next number).
19.6 The formula is thus : the nth number = 8 + (n - 1) x 7
Now calculate the value of the 9th number in the pattern.
19.7 Show that number 22 is greater than 148
19.8 The nth number in the pattern is equal to 78. Calculate the value of n.

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.

 n 1 2 3 4 5 6 Number 65
20.2 Write down the value of a and of d.
20.3 Make a formula to ca;cu;ate a number in this sequence.
20.4 Test your formula by calculating the 5th and 6th numbers in the sequence.
20.5 What is the value of the 9th number?
20.6 The nth number in the sequence is equal to − 1. Calculate the value of n.

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

 n 1 2 3 4 5 6 Number 5
21.2 Write down the value of a.
21.3 The number by which the predecessor is multiplied is called the common ratio and is represented by an r.

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

 n 1 2 3 4 5 Number 1 024
22.2 Write down the value of a and of r.
22.3 Which number is the first number that is smaller than 1? What is the number of its position?

23. Given the formula: b = 3a - 5.
23.1 Write down the relationship between a and b in words.
23.2 Say which is the independent variable and which is the dependent variable.
23.3 Calculate at least 4 values of a and b and write them down in a table (tabulate the results).
23.4 Calculate the value of b if a = –16.
23.5 Calcuate the value of a if b = 4.

24. Given the formula: c = 2 - 3d
24.1 Write down the relationship between a and b in words.
24.2 Say which is the independent variable and which is the dependent variable.
24.3 Calculate at least 4 values of a and b and write them down in a table (tabulate the results).
24.4 Calculate the value of c if d = –6.
24.5 Calculate the value of d if c = 5.

25. Given the formula: p = 6 - 7q
25.1 Write down the relationship between p and q in words.
25.2 Say which is the independent variable and which is the dependent variable.
25.3 Calculate at least 4 values of p and q and tabulate the results.
25.4 Is this an example of an ascending / a rising relationship? Explain.
25.5 Calculate the value of p if q = –2.
25.6 Calculate the value of q if p = -22.

26. Given the formula: y = 2x - 7
26.1 Write down the relationship between x and y in words.
26.2 Say which is the independent variable and which is the dependent variable.
26.3 Calculate at least 4 values of x and y and tabulate the results.
26.4 Is this an example of a descending relationship? Explain.
26.5 Calculate the value of y if x = –3.
26.6 Calculate the value of x if y = 3.

27. Given the formula: xy = 36
27.1 Write down the relationship between p and q in words.
27.2 Say which is the independent variable and which is the dependent variable.
27.3 Calculate at least 4 values of x and y and tabulate the results.
27.4 Are x and y directly proportional to one another? Explain.
27.5 How does the value of y change if the value of x increases / becomes greater in the same proportion? Explain.
27.6 Can x or y ever be equal to zero? Explain.
27.7 Calculate the value of y if x = 3.
27.8 Calculate the value of x if y = 10.

28. Given the formula: pq = 24
28.1 Write down the relationship between p and q in words.
28.2 Say which is the independent variable and which is the dependent variable.
28.3 Calculate at least 4 values of p and q and tabulate the results.
28.4 Are p and q directly proportional to one another? Explain.
28.5 How does the value of q change if the value of p increases / becomes greater in the same proportion? Explain.
28.6 Can x or y ever be equal to zero? Explain.
28.7 Calculate the value of q if p = 12.
28.8 Calculate the value of p if q = 6.

29. Given the formula: ab = 18
29.1 Write down the relationship between p and q in words.
29.2 Say which is the independent variable and which is the dependent variable.
29.3 Calculate at least 4 values of a and b and tabulate the results.
29.4 Are a and b inversely proportional to one another? Explain.
29.5 How does the value of b change if the value of a decreases / becomes smaller in the same proportion? Explain.
29.6 Can x or y ever be equal to zero? Explain.
29.7 Calculate the value of b if a = 12.
29.8 Calculate the value of a if b = 4.
29.9 For which value(s)) is a = b?