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1.
2.
3.
4.
5.
6.
7.
Exploration with patterns
Solving problems by algebraic Method
The Geometer’s Sketchpad
3-dimentional geometry
Statistical investigations
Measures of Central tendency and Spread
Fundamental Counting skills
Reference Book:
Billstein, Libeskind and Lott, A problem solving approach to Mathematics
for elementary school teachers.
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This topic will cover:
• Example of patterns
• Arithmetic sequences
• Geometric sequences
Mathematics has been described as the study of patterns.
Example 1.1
a. Describe any patterns seen in the following:
1+ 0×9 = 1
2 + 1 × 9 = 11
3 + 12 × 9 = 111
4 + 123 × 9 = 1111
5 + 1234 × 9 = 11111
b. Do the above patterns continue? Why or why not?
Solution. a.
(1) It is a sequence of equalities;
(2) Each equality has two terms in the left-hand side and one term in the right-hand
side;
(3) The first term in the left-hand side is a natural number. If this number is r, then the
second term in the left-hand side would be 12L (r − 1) × 9 and the right-hand side
is 11
L
12
31 .
r
b. We can verify that this pattern is also true for r = 6, 7, 8, 9, 10:
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6 + 12345 × 9 = 111111
7 + 123456 × 9 = 1111111
8 + 1234567 × 9 = 11111111
9 + 12345678 × 9 = 111111111
10 + 123456789 × 9 = 1111111111
However, this pattern does not hold if r = 11, as
11 + 12345678910 × 9 = 111111110201.
Hence this pattern does not continue in general.
a. Find four more terms to continue a pattern:
0, 1, 0, 0, 1, 0, 0, 0, 1, ___, ___, ___, ___
b. Describe the patterns found in words.
c. Can you find the number of the 1000th term?
Inductive reasoning:
Inductive reasoning is the method of making generalizations based on
observations and pattern.
Inductive reasoning may lead to a conjecture.
A conjecture is a statement thought to be true but not yet proved true or
false.
For example, considering only that 0 2 = 0 and 12 = 1 , a conjecture might be that
r 2 = r for every natural number r. However, as 2 2 = 4 ≠ 2 , this conjecture does not hold.
A counterexample is an example which shows that a statement does not
holds.
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To show that a conjecture is false, it is enough to exhibit only one counter-example.
Example 1.2. Considering the following
0 3 = 0;
13 = 1
a conjecture might be that r = r for every natural number r.
3
A counterexample to this conjecture: when r = 2, r 3 = 8 ≠ r. Hence this conjecture is
false.
Example 1.3. Considering the following
5 2 = 25;
15 2 = 225;
25 2 = 625;
35 2 = 1225;
we guess this pattern can continue, i.e., the following conjecture can be propose
Conjecture
The following equality holds for all positive integers:
(10r + 5) 2 = r × (r + 1) × 100 + 25 .
We cannot find counterexamples to this conjecture. In fact, we can show that this
conjecture holds for every positive integer r, since
(10r + 5) 2 = (10r ) 2 + 2 × 10r × 5 + 5 2
= 100r 2 + 100r + 25
= 100r (r + 1) + 25
= r × (r + 1 ) × 100 + 25.
As this conjecture holds, we have
45 2 = 2025;
55 2 = 3025;
85 2 = 7225;
L
1. A conjecture is true if it can be shown that it holds for all cases.
2. A conjecture is false if one counterexample to this conjecture can be
found.
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Considering the following
1
1
= 1− ;
1× 2
2
1
1
1
+
= 1− ;
1× 2 2 × 3
3
1
1
1
1
+
+
= 1− ;
1× 2 2 × 3 3 × 4
4
1
1
1
1
1
+
+
+
= 1− ;
1× 2 2 × 3 3 × 4 4 × 5
5
...
(i) describe a pattern;
(ii) make a conjecture based on the pattern found;
(iii) is the conjecture true or false? Why?
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1. A sequence is an ordered arrangement of numbers, figures, or
objects.
2. A sequence has items or terms identified as 1st, 2nd, 3rd, and so on.
Example 1.4 Can you find a property which the first three sequences below have but the
fourth does not?
(a)
(b)
(c)
(d)
1, 2, 3, 4, 5, 6, …,
0, 5, 10, 15, 20, 25, …,
2, 6, 10, 14, 18, 22, …,
1, 11, 111, 1111, 11111, ….
Notice that the first three sequences have the following property:
second term - first term= third term- secondterm= fourth term - third term= ...
However, the fourth sequence does not have this property, as
second term - first term = 11 - 1 = 10;
third term - second term = 111 - 11 = 100;
fourth term - third term = 1111 - 111 = 1000;
L
All the first three sequences are called arithmetic sequences.
An arithmetic sequence is one in which each successive term is obtained
from the previous term by the addition of a fixed number d.
For the first sequence 1, 2, 3, 4, 5, 6, …, the fixed number is d =1, and
2 = 1 + d;
3 = 2 + d;
4 = 3 + d;
...
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For the sequence 0, 5, 10, 15, 20, 25, 30, …, the fixed number is d =5, and
5 = 0 + d;
10 = 5 + d ;
15 = 10 + d ;
...
Exercise 1.1 Which of the following sequences are arithmetic sequences?
(i) 3, 7, 11, 15, 19, …;
(ii) 0, 1, 0, 1, 0, 1, …;
(iii) 2, 5, 8, 12, 16, …;
(iv) 5, 10, 20, 40, ….
Exercise 1.2. Find a pattern in the number of matchsticks required to continue the pattern
shown in the following Figure.
Group 1
group 2
group 3
group 4
For the arithmetic sequence 4, 7, 10, 13, 16, …, every term, except the first term, can be
obtained by adding 3 to its preceding term:
7 = 4 + 3;
Or
10 = 7 + 3;
13 = 10 + 3;
16 = 13 + 3;
...
term (n+1) = term n + 3.
This is an example of recursive pattern.
In a recursive pattern, after one or more consecutive terms are given to start, each
successive term of the sequence is obtained from the previous term(s).
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Now we put the numbers of the sequence 3, 7, 11, 15, 19, … in a table.
Number of term
1
2
3
4
5
.
.
.
n
Term
3
7 = 3+4 = 3+1×4
11= (3+1×4)+4 = 3+2×4
15 = (3+2×4)+4 = 3+3×4
19 = (3+3×4) +4 = 3+4×4
.
.
.
3 + (n-1) ×4
Notice that the nth term of the sequence 3, 7, 11, 15, 19, … is 3 + (n-1) ×4.
In general, we have
Given an arithmetic sequence with the first term a and the difference d between
two consecutive terms, i.e., d = (n+1)th term - nth term for each n, we have
n th term = a + (n − 1)d .
Example 1.5. Assume that in an arithmetic sequence, the second term is 3 and the third
term is 7. Find
(i)
an expression for the nth term;
the 10th term;
(ii)
the 20th term.
(iii)
Solution. Assume that the first term is a and the difference between any two consecutive
terms is d. Then
⎧ a + d = 3;
⎨
⎩a + 2d = 7.
We can find that a = -1 and d = 4. Thus
(i)
The nth term is a+(n-1)d = -1+(n-1)4 = 4n-5;
The 10th term is 4 × 10 − 5 = 35;
(ii)
(iii)
The 20th term is 4 × 20 − 5 = 75.
Exercise 1.3. Assume that in an arithmetic sequence, the fourth term is 13 and the seventh
term is 22. Find
(i)
an expression for the nth term;
the 10th term;
(ii)
The 20th term.
(iii)
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Example 1.6 Find a common property for the following two sequences:
1, 3, 9, 27, 81, …;
(i)
(ii)
3, 6, 12, 24, 48,….
Solution. Both sequences have the property that each term, except the first term, equals to
the product of the preceding term and a constant. For the first sequence,
(n + 1) th term = n th term × 3;
and for the second term, (n + 1) th term = n th term × 2.
Both sequences in Example 1.6 are called geometric sequences.
A sequence is called a geometric sequence if every term, except the first term, can
be obtained from its predecessor by multiplying by a fixed number, the ratio.
Exercise 1.4 Which of the following sequences are geometric sequences?
1, -1, 1, -1, 1, -1, …;
(i)
0, 1, 0, 1, 0, 1, 0, 1, …;
(ii)
1, 2, 3, 4, 5, …;
(iii)
1, 10, 100, 1000, 10000, ….
(iv)
0, 1, 2, 4, 8,16, 32, 64, ….
(v)
1, 3, 2, 9, 4, 27, 8, 81,16, …
(vi)
If a geometric sequence has the first term a and ratio r, then
first term = a;
2 nd term = first term × r = ar ;
3 rd term = 2 nd term × r = ar 2 ;
4 th term = 3 rd term × r = ar 3 ;
L
n th term = ar n −1 .
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Example 1.7. Assume that in a geometric sequence, the second term is 3 and the fourth
term is 12. If all terms are positive, find
an expression for the nth term;
(i)
the 10th term;
(ii)
the 20th term.
(iii)
Solution.
Assume that the first term is a and the ratio is r. Then
⎧ ar = 3;
⎨
3
⎩a × r = 12.
As all terms are positive, we have a = 3/2 and r = 2. Thus
3
(i)
The nth term is ar n −1 = × 2 n −1 = 3 × 2 n − 2.
2
th
10 − 2
The 10 term is 3 × 2
= 3 × 2 8 = 768.
(ii)
th
20 − 2
The 20 term is 3 × 2
(iii)
= 3 × 218.
Exercise 1.5. Assume that in a geometric sequence, the fourth term is 6 and the seventh
term is 162. Find
an expression for the nth term;
(i)
the 10th term.
(ii)
Exercise 1.6. Two bacteria are in a dish. The number of bacteria doubles every hour.
Following this pattern, find the number of bacteria in the dish after 8 hours and after n
hours.
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1.
For each of the following sequences of figures, determine a possible pattern and
draw the next figure according to the pattern:
(a)
(b)
2.
In each of the following, list terms that continue a possible pattern. Which of the
sequences are arithmetic, which are geometric, and which are neither?
(a) 1, 3, 5, 7, 9
(b) 0, 50, 100, 150, 200
(c) 3, 6, 12, 24, 48
(d) 10, 100, 1000, 10000, 100000
(e) 9, 13, 17, 21, 25, 29
(f) 1, 8, 27, 64, 125.
3.
Find the 10th term and the nth term for each of the sequences in Problem 2 above.
4.
Use a traditional clock face to determine the next three terms in the following
sequence:
1, 6, 11, 4, 9, ….
5.
Observe the following pattern:
1 + 3 = 22 ,
1 + 3 + 5 = 32 ,
1 + 3 + 5 + 7 = 42
a. State the next identity according to this pattern;
b. State a generalization based on this pattern.
c. (*) Prove or disprove this generalization.
6.
Each of the following figures is made of small equilateral triangles like the first one.
Following the pattern in the following figures, guess the number of small triangles
needed to make (a) the 100th figure and (b) the nth figure.
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7.
Mak’s first year income is $25,000 and in the 5th year his income is $28,000. If his
income has been increasing each year by the same amount, in which year his income
will be over $40,000 for the first time?
8.
Given the sequence 2,3,5,8,12,17,... , list the next two terms to continue a pattern and
find the 10th term.
9.
Each of the following sequence has some pattern. How many terms are there in each
of the following sequences according to the patterns?
10, 12, 14, 16, 18, …, 2000;
(i)
(ii)
9, 13, 17, 21, 25, …, 209;
(iii) 2, 4, 8, 16, …, 1024.
10. The following sequences are respectively arithmetic sequence and geometric
sequence. Determine the minimum positive integer n such that the nth term of the
geometric sequence is larger than the nth term of the arithmetic sequence:
100, 110, 120, 130, 140, …;
(i)
1, 2, 4, 8, 16, 32, … .
(ii)
11. Find the values of a and b such that the sequence 2, a-b, a+b, 20 is an arithmetic
sequence.
12. Find the values of a and b such that the sequence 2, a-b, a+b, 250 is a geometric
sequence.
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