Thomas Whitham Sixth Form
Algebra
[Type the document subtitle]
S J Cooper
Year 8
Thomaswhitham.pbworks.com
Algebra (1)
Collection of like terms
Simplify each of the following terms:
1.
10m  m  4m
26. 6u  v  u  6v
2.
d  3d  6d  d
27. z  6z  2w  4w
3.
6e  3e  7e
28.  5m  3n  2n  4n  3m
4.
12 f  15 f
29. 7a  b  3c  5a  8b  2c
5.
21x  13x  6x
30. 8e  3 f  5 g  13e  f  4 g
6. 6 y  y  9 y
31. h  9i  7 j  6h  i  3 j
7. 2 p  4 p  3 p
32. 3m  2n  8 p  4m  4n  6 p
8. 7a  4a  9a
33. 9t  7r  s  7r  s  9t
9. 4b  3b  b
34. 5 x  y  8 z  x  4 y  3z
10.  5c  3c  7c
35. 15e  17 f  g  20e  13 f  9 g
11. 8a  3b  a  2b
36. a  2b  a  3b  2a  5b
12. c  5d  4c  3d
37.  3x  2 y  x  y  2 x  4 y
13. 7q  p  3q  4 p
38. 8 p  5q  10r  5 p  2q  7r
14. 9 x  2 y  4 x  8 y
39. 7u  7v  8w  u  10v  13w
15. s  5t  5s  3t
40. 7 x  y  3x  4 x
16. 7e  2 f  6e  f
41. 3t  6u  t  3u
17. 3g  4h  5 g  2h
42. 12a  3b  9a  8b
18. y  2 y  x  7 y
43. x  y  3x  3 y
19. 8m  6n  6m  10n
44. 7 g  5h  g  2h
20. 3i  4 j  6i  3 j
45. 4 j  8k  12 j  k
21.  4 x  6 y  3x  10 y
46. 2 p  9q  12 p  q
22. 5k  7 p  k  4 p
47. 6c  d  5c  3d
23. 9b  7c  8b  3c
48. 8w  3v  2x  7v
24. 8 x  8 y  12 x  6 y
49. 9x  5z  6x  8z
25. 3a  2b  7a  7b
50. a  2b  3c  3a  4b  12c
Algebra (2)
Simplifying Expressions
Simplify each of the following terms:
1.
7 x
16.  5u 11v
2.
3 a  4  b
17. x  x
3.
5 x  3 y
18. p   p
4.
6 c  2 d
19.  2a  9a
5.
7 4 p q
20. 5s  6r
6.
2 y 8 x
21. 4r  r
7.
5  m  7  n
22.  7h  7i
8.
 4  e  5  d
23.  2 p  3q  r
9.
8 f 9 a
24. 6a  2b  3c
10. 3  s  r
25.  e  4 f  7 g
11. 3a  4b
26.  4 x  6 y  z
12.  7m  6n
27. 4t  15t
13.  12q  4 p
28. 6q  p  3r
14. 9k   j
29.  5 y  8 y
15.  7 y  3 x
30.  7m  2n
Algebra (3)
Brackets
1. Remove the brackets for each of the following:
(a) 4x  3
(g) 53x  4
(m)  2x  5
(s) 39  2b
(b) 5x  7
(h) xx  2
(n)  32 x  3
(t) 3mm  2
(c) 3x  1
(i) y2 y  5
(o)  82  3x 
(u) 63  m
(d) 2x  5
(j) 72  3x
(p)  45t  6
(v)  62  3 p 
(e) 22x  3
(k) 41  4 x 
(q)  23b  7
(w)  72a  3b
(f) 32 x  1
(l) 2a3a  7
(r) 2x3  2x
2. Rewrite these expressions using brackets. {Look for common factors}
(a) 2x  8
(f) 4m  6
(k) 3ab  b
(b) 3a  12
(g) 12  9n
(l) 12ef  6e
(c) 3 p  24
(h) 28  21q
(m) 9c  12cd
(d) 8  4x
(i) x 2  3 x
(n) 21xy  12 y
(e) 30  5 y
(j) 3 y 2  2 y
(o) 4 x 2  16 x
3. Remove the brackets and simplify each of the following:
(a) 2x  3  5
(e) x  3  32 x  1
(i) 84 x  5  3x  2
(b) 32 x  4  7
(f) 2x  3  3x  4
(j)
(c) 4  2x  3
(g) 32x  1  22x  5
(d) 4x  22  x
(h) 43x  1  21  3x 
x2 x  1  2 x3  x
Algebra (4)
Simple Equations I
Exercise 1
Find the number which must be placed in the box to make both sides equal.
1.
•+2=5
6.
2+•=6
11.
7-•=5
2.
•+4=6
7.
4+•=7
12.
9-•=6
3.
•+6=9
8.
3+•=6
13.
•-2=3
4.
•+3=7
9.
8-•=6
14.
•-3=5
5.
•+7=9
10.
6-•=2
15.
•-4=5
Exercise 2
Find the number which has to replace the letter to keep both sides equal.
1.
x+3=5
6.
y+4=8
11.
9-p=8
2.
y+3=9
7.
x + 9 = 12
12.
7-y=2
3.
x+5=9
8.
d + 4 = 10
13.
12 - r = 8
4.
x+2=8
9.
2+n=7
14.
y-3=2
5.
x+3=5
10.
1+p=8
15.
6-n=3
Exercise 3
Find the value of the letter in each of the following:
1.
8x = 24
5.
6c = 48
9.
12m = 36
2.
7x = 35
6.
2n = 40
10.
25q = 50
3.
9a = 54
7.
30r = 90
11.
15t = 60
4.
4b = 28
8.
3y = 27
12.
7e = 28
Exercise 4
Find the number which has to replace the letter to keep both sides equal:
1.
2x  3  9
6.
3b  4  16
11.
4q  3  5
2.
2x  6  8
7.
3m  3  18
12.
4r  8  10
3.
2 x  4  12
8.
3n  7  7
13.
6t  2  10
4.
2 y  1  13
9.
2y  4  2
14.
5 x  9  14
5.
3a  2  8
10.
2y  7  3
15.
5v  6  6
Algebra (5)
Solving equations II
Exercise 5
Solve the following equations:
1. 2 x  4  8
2.
2 x  3  13
3.
3 x  2  10
4.
3x  5  4
5.
4 x  1  23
6. 10 x  7  5
7.
2x  3  2
8.
2x  1  7
9.
7 x  4  38
10.
6x  7  5
11. 5  4 x  3
12.
19  7 x  2
13.
68  8 x  12 14.
7  4x  3
15.
20  6 x  7
16. 49  9 x  14
17.
6 x  1  20
18.
2 x  4  10
19.
2 x  3  13
20.
3 x  1  10
21. 3 x  5  11
22.
4x  3  19
23.
10 x  7  57 24.
2x  3  10
25.
2 x  8  13
26. 5 x  7  32
27.
7 x  4  25
28.
23  4 x  7
7  2x  1
30.
68  6 x  8
29.
Exercise 6
Solve the following equations:
1. 2a + 5 = 1
2. 2b + 9 = 3
3. 3c + 4 = 1
4. 5d + 10 = 0
5. 3e + 2 = -4
6. 4x + 7 = -1
7. 6y + 8 = 8
8. 2z + 9 = 5
9. 5s + 3 = -2
10. 7f + 8 = -6
11. 2g + 6 = 5
12. 4h + 3 = -3
13. 3i + 1 = -9
14. 6j + 1 = 7
15. 2m - 2 = 6
16. 2n - 2 = -6
17. 5p + 7 = 7
18. 6q - 12 = 0
19. 5y + 3 = -2
20. 10d + 7 = 57
Exercise 7
Solve the following:
1. 4x + 5 + 3x = 26
2. 7x + 3 + 2x = 21
3. 6y + 12 + 5y = 56
4. 5y + 7 + y = 31
5. a + 9 + 7a = 25
6. 4c - 12 + 2c = 18
7. 9d - 16 + 3d = 20
8. 9q + 12 - 2q = 40
9. 12r + 14 - 6r = 50
10.12r - 6 - 5r = 15
Exercise 8
Solve the following:
1. 3x = 15 - 2x
2. 5x = 16 - 3x
3. 4y = 36 - 5y
4. 6p = 35 - p
5. 3t = 4 - t
6. 5a = 8 + 3a
7. 8b = 18 + 5b
8. 9c = 15 + 4c
9. 5r = r + 32
10.9q = 8q + 6
Exercise 9
Solve the following:
1. 6a + 2 = 2a + 10
2. 9b + 3 = 6b + 18
3. 12c + 9 = 7c + 14
4. 11d + 9 = 4d + 30
5. 5e + 8 = 4e + 15
6. 8p - 7 = 6p + 3
7. 9q - 8 = 3p + 16
8. 11r - 12 = 8r + 6
9. 10u - 20 = 9u - 11
10. v - 8 = 16 - 3v
11. 6x  2  2x  14
12. 10k  4  9k  7
13. 20 y  17  8 y  48
14. 2v  3  45  6v
15. 11s  4  5s  22
16. 15z  8  8z  20
17. 2x  9  19  2x
18. 7r  1  89  3r
19. 4 y  5  41  5 y
20. 7 y  5  3 y  25
Algebra (6)
Solving equations III
Exercise 10
Solve each of the following equations:
1. 2x  3  14
9. 3 y  9  12
17. 5 y  4  10
25. 32  7 x   69
2. 3x  6  18
10. 7x  9  42
18. 7r  8  21
26. 39  2 y   15
3. 3a  4  24
11. 8 y  6  16
19. 42 x  1  36
27. 46  5k   4
4. 2b  5  8
12. 4t  5  24
20. 32q  5  9
28. 85  2x  24
5. 41  x   16
13. 6 p  8  12
21. 53r  4  65
29. 53  2 y   45
6. 6c  3  30
14. 10h  4  50
22. 24 y  1  54
30. 27  3 f   38
7. 5d  5  30
15. 9x  3  63
23. 82m  5  24
8. 2x  7  6
16. 2x  4  8
24. 41  3n  28
Algebra (7)
Construction of equations
1. I think of a number, double it and then add 3. The answer is 15. What was my number?
2. I think of a number, double it and subtract 5. The answer is 7. What was my number?
3. I think of a number, treble it and add 4. The answer is 22. What was my number?
4. I think of a number, multiply it by 3 and subtract 5. The answer is 10. What was my number?
5. I think of a number, multiply it by 4 and subtract 1. The answer is 15. What was my number?
6. I think of a number, multiply it by 7 and add 4. The answer is 39. What was my number?
7. Given the perimeter of the rectangle opposite is 30
x
cm find its length and width
x+3
If the perimeter of the triangle is 45 cm find the sides of the triangle.
8.
2y
2y
y
9. (a) Work out an expression for the perimeter of the rectangle opposite.
20 cm
(b) Given the perimeter is 54 cm find the value of h.
h
10. Given the area of the rectangle above is equal to 80 cm2 work out the value of h.
11. I think of a number, double it and add 5. The result is exactly the same as adding 6 to the
number. What is the number?
12. I think of a number, multiply it by 3 and subtract 5. The result is equal to the same number plus
9. What is the number?
13. I think of a number, multiply it by 3 and add 1. The answer is equal to the same number
doubled add 13. What is the number?
Algebra (8)
Drawing straight line graphs
1. For each of the following
(i)
Copy and complete the table
(ii)
draw the graph for the straight line.
(a)
x
–4
0
4
–3
0
4
–3
0
3
–4
0
4
–4
0
4
y  2x  5
(b)
x
y  3x  2
(c)
x
y  2x  3
(d)
x
y  4x  3
(e)
x
y  5x  3
2.
Draw the graphs for each of the following:
(a)
y  2x  4
(f)
x y 7
(b)
y  3x  7
(g)
x  y  12
(c)
y  5x  8
(h)
x  2y  6
(d)
y  3x  2
(i)
2 x  y  12
(e)
y  7x  4
(j)
y  12 x  3
3. Draw each of the following graphs on the same set of axes. What do you notice?
(a)
y  3x  1
(b)
y  3x  4
(c)
y  3x  5
4. Draw on the same set of axes the graphs of y  2 x  3 and y  3 x . What do you notice?
5. Draw on the same set of axes the graphs of y  2 x  5 and y  4 x  13 . What do you notice?
6. Draw on the same set of axes the graphs of y  3x  2 and x  y  6 . What do you notice?
Algebra (9)
Drawing simple quadratics
1. (a) Copy and complete the table below for the graph of y  x 2  4 for values of x from –3 to 3.
x
–3
–2
–1
0
1
2
3
x2
+4
y
(b) Hence draw the graph of y  x 2  4 .
2. (a) Copy and complete the table below for the graph of y  x 2  4 x for values of x from –3 to 3.
x
–3
–2
–1
0
1
2
3
x2
+4x
y
(b) Hence draw the graph of y  x 2  4 x .
3. (a) Copy and complete the table below for the graph of y  x 2  2 x for values of x from –3 to 3.
x
–3
–2
–1
0
1
2
3
x2
–2x
y
(b) Hence draw the graph of y  x 2  2 x .
4. (a) Copy and complete the table below for the graph of y  x 2  3x for values of x from –4 to 3.
x
–4
–3
–2
x2
+3x
y
(b) Hence draw the graph of y  x 2  3x .
–1
0
1
2
3
5. (a) Copy and complete the table below for the graph of y  x 2  3x for values of x from –4 to 3.
x
–4
–3
–2
–1
0
1
2
3
x2
–3x
y
(b) Hence draw the graph of y  x 2  3x .
6. (a) Copy and complete the table below for the graph of y  x 2  2 x  3 for x from –4 to 3.
x
–4
–3
–2
–1
0
1
2
3
x2
+2x
–3
y
(b) Draw the graph of y  x 2  2 x  3 .
(c) Hence state the values at which y  0 .
7. (a) Copy and complete the table below for the graph of y  x 2  x  6 for x from –4 to 3.
x
–4
–3
–2
–1
0
1
2
3
x2
–x
–6
y
(b) Draw the graph of y  x 2  x  6 .
8. (a) Copy and complete the table below for the graph of y  x 2  2 x  8 for x from –3 to 4.
x
–3
–2
–1
0
1
2
x2
+3x
y
(b) Draw the graph of y  x 2  2 x  8 .
(c) Hence state the coordinates where the curve meets the x-axis.
3
4
Algebra (10)
Simultaneous equations.
1. (i) For each pair of simultaneous equations draw the graphs on the same set of axes.
(ii) Use your graphs to solve the set of simultaneous equations.
(a)
y  2x  3
y  3x  1
(c)
y  3x  1
x  y  13
(b)
y  2x  5
y  4 x  11
(d)
x  2 y  14
x y 6
2. (a) Draw the graph of y  x 2  4 x for values of x between –2 and 4
(b) On the same set of axes draw the straight line with equation y  6 x  3
(c) Hence solve the set of simultaneous equations y  x 2  4 x and y  6 x  3
3. (a) Draw the graph of y  x 2  5 for values of x between –4 and 4
(b) On the same set of axes draw the straight line with equation y  x  2
(c) Hence solve the set of simultaneous equations y  x 2  5 and y  x  2
4. (a) Draw the graph of y  x 2  2 x  1 for values of x between –3 and 5
(b) On the same set of axes draw the graph of y  5 x  3
(c) Hence solve the set of simultaneous equations y  x 2  2 x  1 and y  5 x  3
5. (a) Draw the graph of y  x 2  3x  4 for values of x between –5 and 3
(b) On the same set of axes draw the straight line with equation y  7  5 x
(c) Hence solve the set of simultaneous equations y  x 2  3x  4 and y  7  5 x
6. (a) Draw the graph of y  x 2  2 x  3 for values of x between –4 and 4
(b) On the same set of axes draw the straight line with equation x  y  3
(c) Hence solve the set of simultaneous equations y  x 2  2 x  3 and x  y  3
Algebra (11)
Simultaneous equations I
Exercise 1
Solve each of the following sets of simultaneous equations. Remember to show your working.
1.
x  y  15
x y 7
6.
3x  2 y  7
5x  2 y  1
11.
3a  5b  25
4b  5b  45
2.
a  b  20
a b  4
7.
x  3y  1
2 x  3 y  11
12.
8u  3v  4
2u  3v  14
3.
2x  y  5
3x  y  5
8.
4x  5 y  8
3x  5 y  6
13.
j  4k  20
5 j  4k  4
4.
3 p  q  19
2p  q 1
9.
6x  7 y  4
4 x  7 y  26
14.
7d  2e  29
5d  2e  7
5.
5 x  y  14
3 x  y  10
10.
4 x  3 y  34
x  3y  1
15.
4 p  q  37
pq 3
Exercise 2
Solve each of the following sets of simultaneous equations. Remember to show your working
1.
2.
3.
4.
5.
2 x  y  19
x  y  12
3x  y  8
x y 4
4 p  q  13
2p  q  3
5 x  2 y  24
3 x  2 y  16
5x  3 y  4
2x  3y  7
6.
8a  5b  46
3a  5b  11
11.
9u  7v  50
3u  7v  26
7.
7c  3d  5
2c  3d  10
12.
5m  4n  7
m  4n  3
8.
12 x  5 y  34
6 x  5 y  22
13.
8x  5 y  1
4 x  5 y  13
9.
3g  7h  9
5 g  7h  15
14.
6 j  2k  21
5 j  2k  15
10.
7 x  2 y  19
4 x  2 y  10
15.
9 x  4 y  28
2x  4 y  0
Algebra (12)
Simultaneous equations II
Solve each of the following sets of simultaneous equations. Remember to show your working
1.
2x  y  7
x  2y  5
6.
3 x  4 y  18
5 x  2 y  16
11.
8m  3n  30
4m  2n  16
2.
3x  4 y  5
x  2y  1
7.
7 x  2 y  30
3 x  8 y  20
12.
2x  5 y  8
6x  7 y  8
3.
3x  y  0
2x  3y  7
8.
6 x  5 y  20
2 x  y  12
13.
9e  7 f  25
3e  5 f  1
4.
4a  b  9
5a  4b  14
9.
2 x  3 y  12
4 x  9 y  24
14.
3 x  8 y  21
5x  2 y  1
5.
2 x  3 y  10
5 x  6 y  19
10.
5a  2b  19
7a  8b  24
15.
9 x  10 y  17
3 x  5 y  14
Algebra (13)
Laws of indices
1. Simplify each of the following:
a)
x4  x3
e)
e7  e6  e2
i)
q3  q6  q4  q
b)
p7  p2
f)
h2  h4  h3
j)
y2  y6  y2
c)
a3  a
g)
m  m3  m2
d)
d5 d3 d
h)
x6  x2  x5
2. Simplify each of the following:
a)
a3  b6  a 4  b
e)
p9  q3  p7  q5
i)
e  f 4  e5  f 6
b)
m2  n5  m7  n3
f)
h2  k 3  h5  k 4
j)
u 2  u5  u 7  v3
c)
x4  y 6  y 2  x3
g)
c8  d 2  c  d 3
d)
i5  j3  i4  j 7
h)
m10  n 2  m 2  n 7
3. Simplify each of the following:
a)
x8  x3
e)
b7  b5
i)
t7  t3
b)
y 12  y 5
f)
e4  e3
j)
x13  x13
c)
u7  u6
g)
h 20  h13
d)
p9  p3
h)
d 17  d 15
4. Simplify each of the following:
a)
b)
c)
x 
y 
r 
3 2
d)
2 4
e)
5 3
f)
4a 
3x 
2 p 
5 2
p q 
s t 
2
g)
3 2
4 4
8 3 3
h)
6 5
5. Simplify each of the following expressions:
a)
b)
x7
x5
c)
t 12
t8
d)
c10  c 4
c5
d6 d4
d8
4. Simplify each of the following:
a)
b)
x 
 a b
5 4
2
c)
6
h)
f) p 6 q 2  p 5 q
c d 
a b 
4
3
d)
g) 2r 3 s7  4r 5 s 4
x3 y5
e)
xy 2
7 3
2 7
t   t
t 
u v   v
u v 
6 2
e)
a 8b
4
3 4
5
f)
e6 f 5  e2
f 4  e4 f
4 3
2
4
6
Algebra (15)
Solving Inequalities
1. Solve each of the following inequalities and place each solution onto a separate number line.
a) 2x  1  9
f) 3x  20  11
k) 5x  19  4
b) 3x  2  7
g) 2x  7  5
l) 3x  8  2
c) 2x  7  3
h) 5x  7  3
m) 4x  9  13
d) 3x  1  13
i) 4x  3  7
n) 7 x  8  6
e) 5x  17  13
j) 6x  9  9
o) 3x  8  13
2. Solve each of the following inequalities:
a) 6x  2  2x  14
k)
11e  9  4e  37
11w  12  8w  6
b)
5a  1  2a  8
l)
c)
7 y  5  3 y  25
m) 12 f  9  7 f  14
d)
5g  6  7  4 g
n)
10k  4  9k  7
e)
8e  2  3e  32
o)
h  9  41  4h
f)
3 p  15  2 p
p)
2v  3  45  6v
g)
7r  1  89  3r
q) 11s  4  5s  22
h)
4t  18  2t
r)
15z  8  8z  20
i)
u  8  34  6u
s)
20 y  17  8 y  48
j)
4 y  5  41  5 y
t)
2x  9  19  2x
a) 22 x  1  14
f)
74 x  7  35
b)
32x  1  27
g)
53x  4  35
c)
33x  4  6
h)
27 x  12  4
d)
42 x  9  28
i)
105x  1  140
e)
53x  8  10
j)
87 x  22  8
3. Solve the following inequalities:
Algebra (16)
Substituting into formulae
1. If a  4 p find a when p  7
2. If m  7n find m when n  3
3. Given t  6  s work out t when s  9
4. Given a  c  b what is the value of a when b  14 and c  13 ?
5. If x  y  z what is the value of x when y  34 and z  19 ?
6. If y  2x  1 work out the value of y given x  5 .
7. If y  4x  7 what is the value of y when x  2 ?
8. Given d  6  2e work out the value of d when e  3 .
9. If q  3p  4 what is the value of q when p  12 ?
10. If g  8  4h find the value of g when h  5 .
11. If m  nr  7 work out the value of m when n  8 and r  2 .
12. Given y  mx  c what is the valuw of y when m  2 , x  6 and c  1?
13. Given v  u  at find v when u  7 , a  3 and t  10 .
14.Evaluate S  3T  U when T  9 and U  5 .
15. If V 
M
work out V when M  144 and D  3 .
D
16. Given S 
D
work out S for D  240 and T  12 .
T
17. If h  6i  3 work out h for i  1.
18. Given m  4 n  2 work out the value of m when n  9 .
19. Evaluate P  5 q  r when q  5 and r  2 .
20.Given y  x 2 what is the value of y when x  6 ?
21. Given y  x 2  7 what is the value of y when x  2 ?
22.Given y  x 2  2 x what is the value of y when x  3 ?
23.Given y  x 2  3x what is the value of y when x  3 ?
24.Given y  x 2  x  3 what is the value of y when x  4 ?
25.Given y  x 2  2 x  7 what is the value of y when x  2 ?
Algebra (17)
Continuing Sequences
1. Write down the next two terms for each of the following sequences and a rule in words.
(a)
2
5
8
11
(b)
1
8
15
22
(c)
8
11
14
17
(d)
5
11
17
23
(e)
6
10
14
18
(f)
4
12
20
28
(g)
9
11
13
15
(h)
7
12
17
22
(i)
7
18
29
40
(j)
13
21
29
37
2. Write down the next two terms for each of the following and give a rule for continuing the
sequence.
(a)
20
18
16
14
(b)
37
33
29
25
(c)
99
88
77
66
(d)
47
44
41
38
(e)
56
50
44
38
(f)
80
71
62
53
(g)
63
56
49
42
3. Write down the first five terms for each of the following described sequences.
a) Add 2 to the previous term:
5, …….
b) Add five to the previous term:
9, …….
c) Subtract 1 from the last term:
10, ……
d) Subtract 8 from the previous term:
65, ……
e) Multiply the previous term by 2:
1, …..
f) Multiply the previous term by 3:
3, ……
g) Divide the last term by 2:
64, ……
h) Add the next even number each time:
1, 3, ……
i) Add a number that increases by 2 each time:
3, 4, …….
4. Fill in the missing gaps in the following sequences:
a
3, ... , 17, 24, ..., ...
b
1, ..., 9, 13, ..., ...
c
7, ..., 21, ..., 35, ...
d
8, 11, ..., 17, ...., ...
e
5, ..., 17, ..., 23, ...
f
2, 4, ...., ...., 32, ...
g
4, 10, ..., 22, ...., ...
h
40, ..., 26, 19, ..., ...
i
9, ..., 19, ..., 29, ...
5. Adding together the previous two terms generates the Fibonacci sequence. The first six terms
of the Fibonacci sequence starting with 1, 1 are 1, 1, 2, 3, 5, 8.
(a) Write down the next three terms in this sequence.
(b) Form a new Fibonacci sequence starting with 2, 3 and write down the next five terms.
6. Using the squares in your books draw the next two arrangements for each of the following
patterns:
a
b
c
d
e
f
7. Triangular numbers can be represented by dots arranged as triangles:
Draw and write down the next four in the sequence.
1
3
6
10
8.
Diagram 1
Diagram 2
Diagram 3
Diagram 4
The diagrams above show a series of patterns made up by adding pencils to the previous
diagram.
(a) Draw the next diagram in the sequence.
(b) Copy and complete the table below
Diagram number
1
3
2
5
3
4
5
6
(c) How many pencils will there be in diagram
(i) 7
(ii) 9
Number of pencils
(iii) 15
Algebra (18)
The nth term
1. Give the first four terms of the sequence for which
a) U n  2n  1
e) U n  6n  2
i)
U n  2n
b) U n  3n  4
f) U n  10n  9
j)
U n  n2 1
c) U n  4n
g) U n  n 2
k) U n  5  2n
d) U n  5n  3
h) U n  n 3
l)
U n  n(n  1)
2. The following patterns can be made with matches.
a) Draw the next pattern in the sequence.
b) Copy and complete the following for the first five terms.
Number of small triangles
1
4
9
Number of matches used
3
9
18
c) How many matches will be required for 100 small triangles?
d) How many small triangles can be made using 63 matches?
3. The following is the (incomplete) sequence of natural numbers (positive integers or whole
numbers).
1, 2, 3, 4, 5, 6, ………………
The 3rd term is 3 , the 6th term is 6.
a) Write down the 10th term
b) Write down the nth term.
4. The following is the sequence of even integers 2, 4, 6, 8, 10, 12, …
The 3rd term is 6,
the 6th term is 12.
Write down the values of (a) the 4th term
(b) the 15th term (c) the 28th term (d) the nth term
5. The following is the sequence of odd integers 1, 3, 5, 7, 9, 11, …
The 3rd term is 5; the 6th term is 11.
Write down the values of (a) the 5th term
(b) the 8th term.
To obtain any term, the rule is “double and subtract 1”
(c) Check your answer to (b) [2 x 8 -1 = 15]
(d) Write down the values if (i) the 25th term
(ii) the nth term.
(e) Given the value of a particular term, what is the rule to find which term it is?
6. The following collections of dots suggest another sequence of numbers.
The sequence is started below. Write down the
next seven terms in this sequence.
1,
4,
9,
16,
etc.
a) Write down the values of (i) the 2nd term
(ii) the 10th term.
b) What sorts of numbers belong to this sequence?
c) What is the nth term of this sequence?
d) Which term of this sequence has a value of (i) 64?
(ii) 484?
e) Given the value of a particular term, which button on your calculator would you press to
obtain which term it is?
7. For each of the following sequences write the next two terms in the sequence and then find a
formula for the nth term in the sequence.
a) 1, 3, 5, 7, ….
h) 7, 14, 21, 28, …….
o) 6, 12, 18, 24, …….
b) 2, 8, 14, ….
i) 7, 9, 11, …..
p) 3, 10, 17, ….
c) 4, 7, 10, 13, ……
j) 1, 6, 11, 16, …….
q) 3, 7, 11, ….
d) 1, 7, 13, 19, ………
k) 3, 11, 19, 27, …….
r) 1, 4, 9, 16, 25, …..
e) 5, 9, 13, 17, ……..
l) 9, 18, 27, 36, ……
s) 2, 5, 10, 17, 26, ……
f) 7, 11, 15, …..
m) 5, 12, 19, ……
t) 1, 2, 4, 8, 16, …..
g) 1, 4, 7, 10, ……
n) 1, 0, -1, -2, ………
u) 2 x 3, 3 x 4, 4 x 5, ….
6. Find an expression for the nth term of the sequence
1
2
3
4
5
,
,
,
,
, ......
3
5
7
9 11
Hence obtain the 100th term for this sequence.
7. For each of the following give a formula for the nth term of the sequence
a) 3, 9, 27, 81, …..
b) 0, 3, 8, 15, 24, ….
c) 1 ,
1
1
1
,
,
, ...
4
9 16
8.
This is the beginning of a sequence of tile patterns.
a) Draw the next two patterns
b) Write down the first seven terms of the sequence given by the number of tiles in each pattern.
c) What sort of numbers does the pattern generate?
d) Give the nth term of this sequence.
Algebra (19)
Trial and Improvement
1. (a) Show that x 2  20 has a solution for x between 4 and 5.
(b) Hence using trial and improvement obtain the value of x correct to 1 decimal place.
2. (a) Show that there is a solution for x 2  2 x  18 between x  3 and x  4
(b) Hence using trial and improvement obtain the value of x correct to 2 decimal place.
3. (a) Given x 2  4 x  20 find two consecutive numbers between which the solution lies between.
(b) Hence using trial and improvement obtain the value of x correct to 2 decimal place.
4. (a) Given x 3  20 find two consecutive numbers between which the solution lies between.
(b) Hence using trial and improvement obtain the value of x correct to 1 decimal place.
5. Obtain the value of x, correct to one decimal place given that x 2  4 x  17
6. Given that x 2  3x  23 , find the value of x correct to two decimal places.
7. Find the value of x, correct to 2 decimal places for the equation x 2  7 x  3  56
8. Given x 2 
1
 20 find the values of x correct to 2 decimal places.
x
9. If x 3  x 2  7 obtain a value of x correct to two decimal place.
Algebra (20)
Rearranging formulae
1. Rearrange each of the following so that the letter in the bracket is the subject.
b
y
x
x
 p
v
(a) a  b  3
(b) x  y  7
(c) 2 x  y  3
(d) mx  n
(e) r  p  q
(f) t  v  u
(g) fg  h  0
g 
m
 p
n
m
(i) y  6k
k 
y
x
2
y
(h)
(j)
2. Rearrange each of the following to make x the subject
(a) y  2 x
(f) a  3x  b
(b) y  5 x
(g) d  cx  e
(c) y  x  3
(h) 2 g  3 x  5h
(d) y  2 x  1
(i)
AM  Tx  S
(e) y  4 x  7
(j)
kt  rx  m
3. Rearrange each of the following to make a the subject
(g) 32a  d   k
(a) b  a 2
(b) s  a  2 
2
(h) ma  n  e
(c) g  a 2  h
(i)
p  r e  a 
(d) a 2  b 2  c 2
(j)
a3
b
4
(e) 3a 1  T
(f) 2a  m  n
4. Make y the subject for each of the following:
(a)
y4
x
2
(f)
2q  y 
p
r
(b)
Ty
T
4
(g)
m y  3
a
b
(c)
yM
4
N
(h)
t t  y 
v
u
(d)
yx
x
w
(i)
th
m
y
(e)
ty
a
s
(j)
rs
r
y