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1. Observe the number pattern below.
Line 1:
Line 2:
Line 3:
Line 4: … …
(a) Write down the 5th line of the sequence in a similar pattern. (b) Express the nth line of the sequence in a similar pattern.
(c) Write down two fractions which will give a difference of
124492 .
Express your answer in a similar pattern.
(d) Find the sum of
12+ 1
6+ 1
12+ 1
20+. .. . .. .. .+ 1
24492 without the use of a calculator.
Answer: (a) _______________________________ [1]
(b) _______________________________ [2]
(c) ________________________________ [2]
(d) _______________________________ [2]
2. Study the number pattern in the box below.
Line 1 1 - 2 + 1 = 0Line 2 4 - 4 + 1 = 1Line 3 9 - 6 + 1 = 4Line 4 16 - 8 + 1 = 9
:::
:::
Line p 169 – q + 1 = 144
(a) Complete the 5th and 10th line of this pattern.
Line 5 ……………………………………………………….[1]
Line 10 ………………………………………………………..[1]
(b) Write down the values of p and q.
p = …………………………………[1]
q = …………………………………[1]
(c) Write down the nth line of this pattern
Line n ……………………………………………………….. [2]
3. Write down the next term of the following sequences.
(a) 1, 8, 27, 64, _______
(b)37 ,
613 ,
919 ,
1225 , ________
Ans: (a) __________________ [1]
(b) __________________ [1]
4. Sticks of equal length are used to make a series of patterns. The first 3 patterns are shown below.
(a) Write down the number of sticks needed in the 4th and 10th patterns. [2]
(b) Write down, in terms of n, the number of sticks needed in the nth pattern.
[2]
(c) Explain why 300 sticks cannot be used to form one of the patterns. [1]
5. Consider the patternFirst line: 3 = 1 × 3 = 1 × ( 1 + 2 ) Second line: 8 = 2 × 4 = 2 × ( 2 + 2 )
Third line: 15 = 3 × 5 = 3 × ( 3 + 2 ) .Fourth line: 24 = 4 × 6 = 4 × ( 4 + 2 ) . .
. .
360 = 18 × m = n × ( n + 2 )
(a) Write down the fifth line in the pattern [1]
(b) Find the values of m and n. [1]
(c) Write down the 30th line of the pattern. [1]
6. (a) Write down the next two terms in the sequence
20, 15, 9, 2, ____, _____.
(b) The general term of a sequence is Tn = 6n 18. Find the 9th term.
(c) Write down an expression, in terms of n, for the nth term of the sequence
4, 7, 10, 13, …
Answer (a) ___________ , __________ [1]
(b)_______________________ [1]
(c)_______________________ [1]
7. Consider the sequence Write down(a) the 5th term of the sequence ,(b) an expression for the nth term of the sequence in terms of n ,(c) the 7th term of the sequence .
Answer: (a)The 5th term = __________[2]
(b)The nth term = __________[1]
(c) The 7th term = _________[1]
8. Observe the number pattern below.
Line 1:
Line 2:
Line 3:
Line 4: … …
(a) Write down the 5th line of the sequence in a similar pattern. (b) Express the nth line of the sequence in a similar pattern.
(c) Write down two fractions which will give a difference of
124492 .
Express your answer in a similar pattern.
(d) Find the sum of
12+ 1
6+ 1
12+ 1
20+. .. . .. .. .+ 1
24492 without the use of a calculator.
Answer: (a) _______________________________ [1]
(b) _______________________________ [2]
(c) ________________________________ [2]
(d) _______________________________ [2]
9. The diagram shows a sequence of patterns formed by triangles and dots.
n = 1 n = 2 n = 3
Let the number of lines and dots in the nth pattern be Ln and Dn respectively.
(a) Draw the 4th pattern.
(b) Copy and complete the following table.
n 1 2 3 4
Ln 3
Dn 1
(c) Find an expression for Dn in terms of n.
(d) Find the general term for Ln in terms of Dn.
(e) There are 271 lines in pattern m. Find the value of m.
[1]
[2]
[1]
[1]
[2]
● ●● ● ● ●
10. A series of figures of shaded and unshaded small squares is shown below. The shaded squares are
those that formed a cross-shaped display in the centre. All the other small squares are unshaded.
The table below shows the number of small squares. Figure 1 2 3 4 n
Number of shaded squares
5 9 13 a
Number of unshaded squares
4 12 20 b
(a) Complete Figure 4 column for the
(i) number of shaded squares,
(ii) number of unshaded squares,
(b) Write an equation connecting n and a,
(c) Find the number of small shaded squares in Figure 99.
Answer (a) (i)……….………………….…...[1]
Figure 1 Figure 2
Figure 3
(ii)...……………………….……[1](b) a =……………..…………..……[1]
(c).……...……….…..……………[1]
11. Consider the following pattern
a1 = 3 + 5 = 3 + 1(5) = 8a2 = a1 + 5 = 3 + 2(5) = 13a3 = a2 + 5 = 3 + 3(5) = 18a4 = a3 + 5 = 3 + 4(5) = 23
⋮ ⋮ ⋮an = A = B
(a) Write down the next two lines, a5 and a6, in the pattern.
(b) If ax = 108, find the value of x.
(c) Write down the expressions for A and B in terms of n.
Answer (a) ………………………………………………………… [1]
………………………………………………………… [1]
(b) x = ……………………… [2]
(c) A: ………………………… [1]
B: ………………………… [1]12.
Find the next two terms of the following sequences.
(i) 3, 8, 15, 24,.….., ……,..
(ii) 3, -9, 27, -81,…..., ……,..
Answer : (b)(i) [1]
(ii) [1]
13. The diagram below shows how dots and lines could be fitted to form patterns of hexagons in a row.
(i) Draw the pattern for Fig 4. [1]
(ii) Complete the table which shows the number of bolts needed for Fig1 to Fig 6. [2]
(iii) Find the number of dots needed in Fig 12. [1]
(iv) Write down an expression, in terms of n, for the number of dots needed for
Fig n.
[2]
Fig 1 Fig 2 Fig 3
Figure 1 2 3 4 5
Number of dots 6 10 14
14. The small triangle as shown in figure A is the building block for the rest of the figures to its right. Tn denotes the number of building blocks in the nth figure in the sequence. For example, T2 = 4 and T3 = 9.
T1 T2 T3 T4
(a) Write down T4.(b) Find the general term Tn in terms of n.(c) Which figure in the sequence will have 256 triangular building blocks in it?
Answer: (a) _______________ [1]
(b) _______________ [1]
(c) _______________ [2]
15 Complete the following number patterns.
(a) 54, 50, 47, 45, 44, 44, 45, _____, _____.
(b) 5, 6, 10, 19, 35, 60, _____, _____.
Figure A
Answer (a) , [2]
(b) , [2]Study the pattern shows below:
Answer (a) m = [1]
(b) x = [1]
Fig. 1 Fig. 2 Fig. 3 Fig. m Fig. 12
y z
x
36 37
35
… 9 10
8
4 5
3
1 2
0
16.
(a) Find the value of m in Figure m.
(b) Calculate the values of x, y and z in Figure 12.
y = [1]
z = [1]
18. (a) Write down the next two terms in the sequence:
2, 5, 8, 11, 14, _____, _____
(b) The nth term of a sequence of numbers is
n×(n+1)2 , wheren≥1 .
Write down the first four terms of this sequence.
17. Consider the following pattern.
(a) Write down, , the 7th line in the pattern. [1]
(b) Write down, , the nth line in the pattern. [3]
(c)If , find the value of w.
[1]
Answer (a) ……………….. ,…………….. [1]
(b) .........., ………, ………, …….. [2]
19. The diagram below shows the first three sequences of dot patterns.
1st 2nd 3rd
(a) Draw the 4th pattern. [1]
(b) The information from the sequence of dots is tabulated below.
Copy and complete the table. [2]
Pattern Formula Number of Dots
1 1 1
2 4 + 1 5
3 4 + 4 + 1 9
4
(c) Write down a formula to calculate the number of dots in the nth pattern. [1]
(d) Hence, find the number of dots in the 20th pattern. [1]
20. Consider the number pattern below :
3, 8, 13, 18, 23, 28, …
(a) Write down the 9th term of the pattern. [1]
(b) Write down the nth term of the pattern. [1]
(c) Write down the term which has a value of 188. [1]
(d) Hence, find, without the use of a calculator, the sum of
3 + 8 + 13 + 18 + 23 + 38 + ….+ 188 [2]
21. The diagram below shows the first three of a sequence of dot patterns.
a. Draw the 4th pattern and hence, write down the number of dots in the 4th pattern. [1]b. The information from the sequence of dots is tabulated below.
Complete the table. [2]
Pattern Formula Number of dots1 1 12 4 1 53 4 4 1 945
c. Write down a formula to calculate the number of dots in the nth pattern. [1]d. Hence, find the number of dots in the 25th pattern. [1]
22. The diagrams below show a sequence of squares stacked up in the following pattern
n = 1 n = 2 n = 3
(a) Draw the 4th diagram [1]
(b) Complete the following table where Tn is the number of squares in the nth diagram.[1]
(c) Express Tn in terms of n.(d) If the kth diagram has 225 squares, find the value of k.
n 1 2 3 4 5
Tn 1 4 9
Answer: (c) ________________ [1]
(d) ________________ [1]
23. Observe the number pattern below.
32−12=4 (3−1 )=8 (1 )=852−32=4(5−1)=8 (2)=1672−52=4(7−1)=8(3 )=2492−72=4( 9−1 )=8( 4 )=32 ⋮ ⋮ ⋮ ⋮
Line n p2 – q2 = 4(p – 1) = 8n= 160
(a) Write down the next line of the number pattern.
(b) Find the values n, p and q.
(c) Write down an expression for p in terms of n.
Answer: (a) ___________________________________________________ [1]
Line 1Line 2Line 3Line 4
(b) ____________________ [3]
(c) ____________________ [1]
24. For each of the following sequences, write down the two missing terms.
(a)
(b)
1, 3, 6, 10, ______, ______.
______, ______, 120, 60, 30, 15.
Answer (a) ................. , ................... [1]
(b) ................. , ................... [1]
25. Look at the following figures that are formed by squares of the same size.
Figure 1 Figure 2 Figure 3
(a) Complete the table below:
Figure number Number of squares
1 1
2 5
3
4
[2]
(b) How many squares are there in Figure 20? [1]
(c) Write down the number of squares, in terms of n, in Figure n. [1]
26. The first four terms of a number sequence are 5, 12, 19, 26, ... .
(a) Find the next three terms of the sequence.
(b) Hence, write down the general term, T n of this sequence.
Answer: (a) __________________ [2]
(b) __________________ [2]
27. Find the 6th term in each of the following sequences.
(i) 2 , 0 , – 2 , – 4 , – 6, … [1]
(ii) 0 , 3 , 8 , 15 , 24 , … [1]
28.
In a restaurant, 4 chairs will be placed around a square table. If there are more customers, more chairs and tables will be added as shown below.
Pattern 1 Pattern 2 Pattern 3 Pattern 4
(a) Draw the structure for Pattern 4 within the box given above. [1](b) Suppose each side of the table is 1.2 m and metre is the perimeter of the th figure.
(i) Find , and
(ii) State the general term (c) Find the number of chairs and tables required for Pattern 10.
Answer (b)(i) = [1]
= [1]
= [1]
(b)(ii) = [1]
(c) No of chairs = [1]
No of tables = [1]
29. The number pattern consists of dots, lines and small equilateral triangles of sides 1 cm long.
Figure
n Figure 1 Figure 2 Figure 3 Figure 4
Number of dots 3 6 10 e
Number of small triangles 1 4 9 16
Perimeter of figure 3 cm 6 cm 9 cm f
By considering the number patterns in the table above,
(i) find the values of e and f. [2]
For the Figure n, find in terms of n, an expression for
(ii) the number of small triangles and [1]
(iii) the perimeter of each figure. [1]
