Sequence of Numbers. Mun Chou, Fong QED Education Scientific Malaysia

Size: px

Start display at page:

Download "Sequence of Numbers. Mun Chou, Fong QED Education Scientific Malaysia"

Nickolas Singleton
7 years ago
Views:

1 Sequence of Numbers Mun Chou, Fong QED Education Scientific Malaysia LEVEL High school after students have learned sequence. OBJECTIVES To review sequences and generate sequences using scientific calculator. OVERVIEW We briefly review properties of sequence of numbers; following the review we explore about using scientific calculator to generate the sequence from different approaches. In this exploration the calculator in discussion is the fx-991es. EXPLORATORY ACTIVITIES [Note] (a) We shall use small letter x an y instead of capitals X and Y as shown on the calculator throughout the paper. (b) Unless otherwise specified, we choose MthIO mode in the SETUP menu by tapping Sequence in Review Sequence is a function by definition. We usually deal either with finite or infinite sequence. For example, the sequence of numbers 1,, 5, 7, 9 is a finite sequence; while the function f ( n) = n + 1, for n positive integer set, is an infinite sequence. Some other finite and infinite sequences: (i) Prime numbers:,, 5, 7, 11, 1, 17, is an infinite sequence. (ii) All positive odd number less then 0: 1,, 5, 7, 9, 11, 1, 15, 17, 19 is finite (iii) Fibonacci numbers: 1, 1,,, 5, 8, 1, 1, 4, is infinite. Usually, the terms within the sequence can be described by some pattern and as such the sequence can be represented with a general term, although some sequences do not 005 CASIO Computer Co., LTD. All Right Reserved. Page A

2 observe any general patterns, such as the sequence of prime numbers. In this exploration we denote the general term as T n. In the study of sequence we are usually required to do one or all of the following: generate the sequence, find the general term, find the n-th term and calculate the sum of the first n-th term. For illustration let s consider this example. Example 1: Given the sequence of all positive even numbers, (i) Write down the first 7 terms of the sequence. To generate the infinite sequence is fairly easy:, 4, 6, 8, 10, 1, 14, (ii) Find the general term of the sequence. The general term can be represented as such: T n = n, where n positive integer set. (iii) Find the 15 th term. Using the general term, we have T 15 = (15) = 0. (iv) Find the sum of the first 1 term. The sum of the first n-th term is denoted as S n in this exploration. We are required to find S 1 = = 156. As this example demonstrated, some calculations can be quite tedious. We now explore the use of calculators to speed up the calculations. We also explore the possibility of using the calculator to find the n-th term and the sum, with the general term T n given. Generate the Sequence with Table Using the calculator we can generate the sequence much more efficiently. One good way is by the use of Table. Example : The n-th term of a sequence is T n =. Write down the first 15 terms. Using the Table mode, we can generate these 15 terms efficiently. The Start value is 1 and End value is 15 while step size is always 1 for sequence. In the calculator we use x to replace the n given in the general term. Enter Table mode x( x + 1) Now input f ( x) = x( x + 1) Continue to input f ( x) = Enter the Start and End values Followed by the Step size A table of values will be tabulated from this function and the values on the F(X) column are the first 15 terms for the sequence represented by T n =. 005 CASIO Computer Co., LTD. All Right Reserved. Page A

3 We can also find the n-th term using the table if we input the setting such that n is Start value n End value of the table. Sequence with Solver A much better approach if we are to find the n-th term is through the use of the calculator solve features as this feature is more accessible than the Table mode. Example : Find the arithmetic mean of the 9 th, 49 th and the 7 rd with the general term T n = n 4n + 8. terms of the sequence T9 + T49 + T7 We are required to find and therefore the solution to this problem is not too hard but the calculation is tedious. With the combination of the CALC function and the independent memory, the problem can be calculated quite efficiently. Here again we replace n in the general term with x. Clear the independent memory Now input x 4x + 8 Continue to input x 4x + 8 Now calculate T 9 This gives T 9 = Now store this value into the independent memory M. Store T 9 into M Now scroll up to find T 49 Add T 49 to T 9 stored in M Scroll up to find T 7 Add T 7 to sum in M 9 + T49 + T Now find T Hence the arithmetic mean is. This example shows that proper use of technology such as a scientific calculator can help to approach a problem efficiently and effectively. 005 CASIO Computer Co., LTD. All Right Reserved. Page A

4 Recursive Sequence with Answer Memory Sequence where its n-th term is the function of previous terms is a recursive sequence. The first term of a recursive sequence is usually given. We look at how we may utilize the Answer Memory, or Ans, to help us solve problems on recursive sequence. Example 4: The first term of the recursive sequence Tn = Tn 1 + Tn 1 is given as. Find the first 5 terms of the sequence. From the calculation perspective the generation of the 5 terms is not difficult but tedious. We can improve the calculation using the Answer Memory as follow. We key in the expression Tn = Tn 1 + Tn 1 but with the (n-1)-th term replaced by Ans. Store into the Answer Memory Now enter Ans + Ans Now calculate T T 1 is given as in the problem, and our first press of actually calculates () + () = 10, and this is the second term T. We can find the next terms by simply tapping on the equal key. To calculate T To calculate T 4 To calculate T 5 The operation shows that the first 5 term of the recursive sequence as, 10, 10, 1790, Sum of the First n-th Terms Finding sum of the first n-th term, or S n, is important in the study of sequence. While calculating the sum for a recursive sequence and sequence which has no general pattern demands further study, finding the sum is usually quite academic though again the working involved is tedious. The Summation feature of the 991-ES provides us with avenue to improve this. Example 5: Find the sum of the first terms, or S, for the following sequences. (i) The sequence of numbers with general term T n = n -1, for n = 1,,, 4, (ii) The sequence described in Example whose general term is T n =. 005 CASIO Computer Co., LTD. All Right Reserved. Page A

5 The Summation feature () x = requires us to input the start and end values for the summation. The step size, or increment, within the Summation is fixed as 1. Hence in using the Summation feature to solve (i), the operation is as follow. Enter the Summation mode Move cursor to start value box Enter 1 when the box is selected Now move up to end value box Enter into the end value box Scroll to the function term box Using x to replace n in T n = n -1, enter the general term. Enter x -1 To evaluate The result is 59, and apparently the sequence is the sequence of positive odd number. This result means = 59. To solve (ii) the operation is as followed. Enter the Summation mode Move cursor to start value box Enter 1 when the box is selected Now move up to end value box Enter into the end value box Scroll to the function term box Enter Continue entering then evaluate The result is 4550 as shown on the calculator. The summations of sequence, or S n, are usually referred to as Series. 005 CASIO Computer Co., LTD. All Right Reserved. Page A

6 EXERCISES Use the few features discussed above to solve these exercises. You can choose to work on the same problems with different features or a combination of them. Exercise 1 Write down the 5 th till 19 th term of the sequence with general term T n = 5 n. Exercise Find the first 14 terms of the sequence with general term T n = calculate the sum of its first 14 terms. 7n n n +. Then Exercise Find the geometric mean of the 5 th, 7 th and the 10 th terms of the sequence with the general term 5n T n = n + 1 Hint: Geometric mean of numbers α, β and γ is αβγ. Exercise 4 The first term of the recursive sequence T = n T n 1 1 is given as 5. Find the sum of the first 7 terms of the sequence. Exercise 5 Find the sum of the first 100 terms for the sequences (i) T n = n 1 and (ii) T n = n. SOLUTIONS to Exercises Exercise 1-45, -67, -9, -1, -157, -195, -7, -8, -, -87, -445, 507, -57, -64, -717 Exercise 6, 50, 174, 40, 80, 1446, 10, 464, 4950, 6810, 9086, 1180, 15054, Sum =7550. Exercise Geometric mean is correct to 6 decimal places. Exercise 4 Approximately.6107 X Exercise 5 (i) 850 and (ii) CASIO Computer Co., LTD. All Right Reserved. Page A

4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences