. Exploration with patterns. Solving problems by algebraic Method. The Geometer s Sketchpad 4. -dimentional geometry 5. Statistical investigations 6. Measures of Central tendency and Spread 7. Fundamental Counting skills Reference Book: Billstein, Libeskind and Lott, A problem solving approach to Mathematics for elementary school teachers. -
This topic will cover: Example of patterns Arithmetic sequences Geometric sequences Mathematics has been described as the study of patterns. Example. a. Describe any patterns seen in the following: + 0 9 = + 9 = + 9 = 4 + 9 = 5 + 4 9 = b. Do the above patterns continue? Why or why not? Solution. a. () It is a sequence of equalities; () Each equality has two terms in the left-hand side and one term in the right-hand side; () 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 L ( r ) 9 and the right-hand side is L. r b. We can verify that this pattern is also true for r = 6, 7, 8, 9, 0: -
6 + 45 9 = 7 + 456 9 = 8 + 4567 9 = 9 + 45678 9 = 0 + 456789 9 = However, this pattern does not hold if r =, as + 4567890 9 = 00. Hence this pattern does not continue in general. a. Find four more terms to continue a pattern: 0,, 0, 0,, 0, 0, 0,,,,, b. Describe the patterns found in words. c. Can you find the number of the 000 th 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 = 0 and =, a conjecture might be that r = r for every natural number r. However, as = 4, this conjecture does not hold. A counterexample is an example which shows that a statement does not holds. -
To show that a conjecture is false, it is enough to exhibit only one counter-example. Example.. Considering the following 0 = 0; = a conjecture might be that r = r for every natural number r. A counterexample to this conjecture: when r =, r = 8 r. Hence this conjecture is false. Example.. Considering the following 5 5 5 = 5; = 5; = 65; 5 = 5; we guess this pattern can continue, i.e., the following conjecture can be propose Conjecture The following equality holds for all positive integers: (0r + 5) = r ( r + ) 00 + 5. We cannot find counterexamples to this conjecture. In fact, we can show that this conjecture holds for every positive integer r, since (0r + 5) = (0r) + 0r 5 + 5 = 00r + 00r + 5 = 00r( r + ) + 5 = r (r + ) 00 + 5. As this conjecture holds, we have 45 55 85 L = 05; = 05; = 75;. A conjecture is true if it can be shown that it holds for all cases.. A conjecture is false if one counterexample to this conjecture can be found. -4
-5 Considering the following... ; 5 5 4 4 ; 4 4 ; ; = + + + = + + = + = (i) describe a pattern; (ii) make a conjecture based on the pattern found; (iii) is the conjecture true or false? Why?
. A sequence is an ordered arrangement of numbers, figures, or objects.. A sequence has items or terms identified as st, nd, rd, and so on. Example.4 Can you find a property which the first three sequences below have but the fourth does not? (a),,, 4, 5, 6,, (b) 0, 5, 0, 5, 0, 5,, (c), 6, 0, 4, 8,,, (d),,,,,. Notice that the first three sequences have the following property: s econd term - first term= thirdterm- secondterm= fourth term- thirdterm=... However, the fourth sequence does not have this property, as second term - first term = - = 0; third term - second term = - = 00; fourth term - third term = - = 000; 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,,, 4, 5, 6,, the fixed number is d =, and = + d; = + d; 4 = + d;... -6
For the sequence 0, 5, 0, 5, 0, 5, 0,, the fixed number is d =5, and 5 = 0 + d; 0 = 5 + d; 5 = 0 + d;... Exercise. Which of the following sequences are arithmetic sequences? (i), 7,, 5, 9, ; (ii) 0,, 0,, 0,, ; (iii), 5, 8,, 6, ; (iv) 5, 0, 0, 40,. Exercise.. Find a pattern in the number of matchsticks required to continue the pattern shown in the following Figure. Group group group group 4 For the arithmetic sequence 4, 7, 0,, 6,, every term, except the first term, can be obtained by adding to its preceding term: 7 = 4 + ; 0 = 7 + ; = 0 + ; 6 = + ;... Or term (n+) = term n +. 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). -7
Now we put the numbers of the sequence, 7,, 5, 9, in a table. Number of term Term 7 = +4 = +4 = (+4)+4 = +4 4 5 = (+4)+4 = +4 5 9 = (+4) +4 = +44...... n + (n-) 4 Notice that the n th term of the sequence, 7,, 5, 9, is + (n-) 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+) th term - n th term for each n, we have n th term = a + ( n ) d. Example.5. Assume that in an arithmetic sequence, the second term is and the third term is 7. Find (i) an expression for the n th term; (ii) the 0 th term; (iii) the 0 th term. Solution. Assume that the first term is a and the difference between any two consecutive terms is d. Then a + d = ; a + d = 7. We can find that a = - and d = 4. Thus (i) The n th term is a+(n-)d = -+(n-)4 = 4n-5; (ii) The 0 th term is 4 0 5 = 5; (iii) The 0 th term is 4 0 5 = 75. Exercise.. Assume that in an arithmetic sequence, the fourth term is and the seventh term is. Find (i) an expression for the n th term; (ii) the 0 th term; (iii) The 0 th term. -8
Example.6 Find a common property for the following two sequences: (i),, 9, 7, 8, ; (ii), 6,, 4, 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, th th ( n + ) term = n term ; th th and for the second term, ( n + ) term = n term. Both sequences in Example.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.4 Which of the following sequences are geometric sequences? (i), -,, -,, -, ; (ii) 0,, 0,, 0,, 0,, ; (iii),,, 4, 5, ; (iv), 0, 00, 000, 0000,. (v) 0,,, 4, 8,6,, 64,. (vi),,, 9, 4, 7, 8, 8,6, If a geometric sequence has the first term a and ratio r, then first term = a; nd rd term = first term r = ar; term = nd term r = ar th 4 rd term = term r = ar ; L th n n term = ar. ; -9
Example.7. Assume that in a geometric sequence, the second term is and the fourth term is. If all terms are positive, find (i) an expression for the n th term; (ii) the 0 th term; (iii) the 0 th term. Solution. Assume that the first term is a and the ratio is r. Then ar = ; a r =. As all terms are positive, we have a = / and r =. Thus (i) The n th n n n term is ar = =. (ii) The 0 th 0 8 term is = = 768. (iii) The 0 th 0 8 term is =. Exercise.5. Assume that in a geometric sequence, the fourth term is 6 and the seventh term is 6. Find (i) an expression for the n th term; (ii) the 0 th term. Exercise.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. -0
. For each of the following sequences of figures, determine a possible pattern and draw the next figure according to the pattern: (a) (b). 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),, 5, 7, 9 (b) 0, 50, 00, 50, 00 (c), 6,, 4, 48 (d) 0, 00, 000, 0000, 00000 (e) 9,, 7,, 5, 9 (f), 8, 7, 64, 5.. Find the 0 th term and the n th term for each of the sequences in Problem above. 4. Use a traditional clock face to determine the next three terms in the following sequence:, 6,, 4, 9,. 5. Observe the following pattern: + = + + 5 + 7 = 4 + + 5 =,, 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 00 th figure and (b) the n th figure. -
7. Mak s first year income is $5,000 and in the 5 th year his income is $8,000. 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,,5,8,,7,..., list the next two terms to continue a pattern and find the 0 th 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? (i) 0,, 4, 6, 8,, 000; (ii) 9,, 7,, 5,, 09; (iii), 4, 8, 6,, 04. 0. The following sequences are respectively arithmetic sequence and geometric sequence. Determine the minimum positive integer n such that the n th term of the geometric sequence is larger than the n th term of the arithmetic sequence: (i) 00, 0, 0, 0, 40, ; (ii),, 4, 8, 6,,.. Find the values of a and b such that the sequence, a-b, a+b, 0 is an arithmetic sequence.. Find the values of a and b such that the sequence, a-b, a+b, 50 is a geometric sequence. -