Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
.. { Lesson 1.4 Finding the nth Term ExampleB (jot! trUU4 tfu inUlJers, aff we i 'UI01t of mono Tenn Value - £eopoM ~rone The pattern of numbers 3, 4, 5, 6, 7. 8, 9, ... is an example of a sequence of numbers. The first term of the sequence has a vaI"ueof three, the second term has a value of four, the third term has a value of five, and so on. Let's show this as a table. Term 1 2 3 4 5 6 .. 20 Value 3 4 5 6 7 8 ... 22 , ... ... 200 ... 202 ... n+2 3 4 5 6 -2 -1 0 1 2 3 ... ... 20 -?- ... ... 200 -1- ... ... 5 6 .... 20 ... 200 ... n 2 4 6 8 10 12 ... -?- ... -1- ... -7- Term 1 Value 1 2 4 3 3 5 7 5 6 9 11 ... ... 20 -?- '" 200 ... ... -'!- ... -?- ·n Every odd number is one less than an even number. 1,3,5.7,9, ... are each one less than a term in the even sequence 2,4,6,8, 10, .... Therefore, the value of each term of the sequence of odd nwnbers is one less than twice the number of the term. The value of the 20th term is 2(20) - 1 or 39. The value of the 200th term is 2(200) - I or 399. In general, the value of the nth term is 2n - 1. EXERCISE SET A Find the nth term in ea.h sequence. 1-;* n 2.. 1 2 3 4 5 6 ... 5 6 7 8 9 10 ... -?- n 3. -?- To find the rule or formula that describes the sequence above, notice that the value of each tennis three less than the number of the term, Therefore, the 20th term of the sequence will be 20 .; 3 or 17. The value of the 200th tenn will be 197. In general, the value of the nth term is n - 3. 50 I Chapter 1 ( 4 ExBmpleC'" Example A 2 3 SometimeS the patterns are created by a combination of addition 01' subtraction with multiplication. The pattern beloW is the familiar sequence of odd numbers. n In previous lessons you found the next term of many sequences, What if you needed to know the value of the 20th, 200th, or 2000th term of a number sequence but you didn't know the rule? You certainly don't want to calculate all the values in between just to get one answer. What you need is a formula or rule that describes the sequence of values. Let's look: at a few examples to See how you find these rules. The simplest of the formulas involve simple addition, subtraction, or multiplication by an integer. 1 2 To find the rule or formula that descnl:Jes the sequence above, notice that the value of each term is twice the value of the nwnber of the term. The value of the 20th term of the sequence is 2(20) or 40. The value of the 200tb term is 400. In general, the value of the nth term is 211. This pattern is the sequence of even numbers. Each number in the top row represents a term or position number in the sequence. The number beneath each term number is the value of that term of the sequence. The n and the n + 2 tell us the rule that connects the top row with the bottom row. This particular rule states that to find the value of any particular term in this sequence, you simply add two to the term number. Therefore, if you wish to know the value of the twentieth term oftbis sequence, you add 20 + 2 to get 22. The value of the twentieth term of this sequence is 22. Term Value 1 1 2 3 4 5 6 ... 0 1 2 3 4 5 ... -?- 1 2 3 4 5 6 4 7 10 13 16 19 ... ... n 4. 4 1 2 3 5 6 4 8 12 16 20 24 ... n ... -?- n -?- Inductive Reasoning I 51 ( ( (D .. { Lesson 1.4 Finding the nth Term ExampleB (jot! trUU4 tfu inUlJers, aff we i 'UI01t of mono Tenn Value - £eopoM ~rone The pattern of numbers 3, 4, 5, 6, 7. 8, 9, ... is an example of a sequence of numbers. The first term of the sequence has a vaI"ueof three, the second term has a value of four, the third term has a value of five, and so on. Let's show this as a table. Term 1 2 3 4 5 6 .. 20 Value 3 4 5 6 7 8 ... 22 , ... ... 200 ... 202 ... n+2 3 4 5 6 -2 -1 0 1 2 3 ... ... 20 -?- ... ... 200 -1- ... ... 5 6 .... 20 ... 200 ... n 2 4 6 8 10 12 ... -?- ... -1- ... -7- Term 1 Value 1 2 4 3 3 5 7 5 6 9 11 ... ... 20 -?- '" 200 ... ... -'!- ... -?- ·n Every odd number is one less than an even number. 1,3,5.7,9, ... are each one less than a term in the even sequence 2,4,6,8, 10, .... Therefore, the value of each term of the sequence of odd nwnbers is one less than twice the number of the term. The value of the 20th term is 2(20) - 1 or 39. The value of the 200th term is 2(200) - I or 399. In general, the value of the nth term is 2n - 1. EXERCISE SET A Find the nth term in ea.h sequence. 1-;* n 2.. 1 2 3 4 5 6 ... 5 6 7 8 9 10 ... -?- n 3. -?- To find the rule or formula that describes the sequence above, notice that the value of each tennis three less than the number of the term, Therefore, the 20th term of the sequence will be 20 .; 3 or 17. The value of the 200th tenn will be 197. In general, the value of the nth term is n - 3. 50 I Chapter 1 ( 4 ExBmpleC'" Example A 2 3 SometimeS the patterns are created by a combination of addition 01' subtraction with multiplication. The pattern beloW is the familiar sequence of odd numbers. n In previous lessons you found the next term of many sequences, What if you needed to know the value of the 20th, 200th, or 2000th term of a number sequence but you didn't know the rule? You certainly don't want to calculate all the values in between just to get one answer. What you need is a formula or rule that describes the sequence of values. Let's look: at a few examples to See how you find these rules. The simplest of the formulas involve simple addition, subtraction, or multiplication by an integer. 1 2 To find the rule or formula that descnl:Jes the sequence above, notice that the value of each term is twice the value of the nwnber of the term. The value of the 20th term of the sequence is 2(20) or 40. The value of the 200tb term is 400. In general, the value of the nth term is 211. This pattern is the sequence of even numbers. Each number in the top row represents a term or position number in the sequence. The number beneath each term number is the value of that term of the sequence. The n and the n + 2 tell us the rule that connects the top row with the bottom row. This particular rule states that to find the value of any particular term in this sequence, you simply add two to the term number. Therefore, if you wish to know the value of the twentieth term oftbis sequence, you add 20 + 2 to get 22. The value of the twentieth term of this sequence is 22. Term Value 1 1 2 3 4 5 6 ... 0 1 2 3 4 5 ... -?- 1 2 3 4 5 6 4 7 10 13 16 19 ... ... n 4. 4 1 2 3 5 6 4 8 12 16 20 24 ... n ... -?- n -?- Inductive Reasoning I 51 ( ( (D ( ( ( Lesson 1.5 ('\ The triangular number sequence doesn't always begin with a one. What would be the rule for the triangular number sequence: 6, 10, 15, 21,28, ... '1 Let's look. Triangular Numbers ~.' .' ExampleS • •• • •••3 ••• 6 • •• • ••••• 10 15 ••• ••••• •••• •••• • •• ••• •••• ••••• •••••• 21 The sequence above appears in many different geometry problems. The sequence is called the triangular numbers. The ancient Greeks were probably the first to work with the triangular number sequence. Let's look at how you can find the rule for this sequence of triangular numbers. Example A In this pattern it is easy to get the next term in the sequence, but not so easy to find the rule that describes the term. You can try factoring 1,3,6,10,15,21, hoping for a pattern to emerge that relates the terms to the factors. Value 1 1 Factored 1- 1 Term ·2 3 4 5 6 3 6 10 15 21 1- 3 1• 6 2·3 ... 200 ... n ... -7- ... -7- ... -?- ... 20 2 3 4 5 6 Value 1 3· 6 10 15 21 Doubled 2 6 12 20 30 42 2 3 4 5 6 Value 6 10 15 21 28 36 Doubled 12 20 30 42 56 72 ED ... 20 ... 200 ... n ... -?- ... -7- ... -7- I· 12 1- 20 1- 30 1· 42 2·6 2-10 2·15 2·21 3-10 3-14 90 Since the nth term of the doubled sequence is n(n + 1), then half of that gives the nth term of the triangular number sequence. The nth triangular number is 54 I Chapter 1 €I) EXERCISE SET A n('!. + 1) ( 2. 1 2 3 4 5 6 0 1 3 6 10 15 ... ... ... n n 1 2 3 4 5 6 -7- 0 2 5 9 14 20 1 2 3 4 5 6 .. , 2 5 9 14 20 27 ... -7- 1 2 3 6 ... n ... -7- , .. -'1- 4. 3. 1 2 3 4 3 6 10 15 21 5 ... n 28 ... -76 n 6.*- 5. -1 e·ee 1- 6 20 sa 1 Factored ... 200 ... n ... -1- ... -7- .., -1- .. , Factored 1 - 12 1 - 20 1 - 30 1- 42 1- 42 1- 72 2 - 6 2 - 10 2· 15 2 - 21 2 - 28 2 - 36 3 - 10 3 - 14 4 - 14 3·24 . ~cr:D4'18 6·12 1.* But this doesn't seem to be too helpful. There is a pattern, but it appears to alternate (one pattern with odd terms end another with even terms). H factoring doesn't help, it may be because the formula involves division as well as factoring. If you suspect that the rule involves division, try doubling or tripling the values, and then factor these numbers. If the doubled values give you a set of numbers that can be more readily put into a formula, then this formula must be divided in half to get the formula for your original sequence. This works with triangular numbers. 1 1 Find the nth term in each sequence. 1· 10 1· 15 1- 21 2·5 3-5 3·7 Term To find the rule that describes the triangular number sequence below, you would probably first look at the factors, but you would again have no luck. If you suspect that the rule involves division, you should try doubling or tripling the values, and then factor these numbers. The doubled values will again give you a set of numbers that can be more readily put into a formula. This formula must be divided in half to get the formula for your originaIsequence. Tenn 2 -1 3 0 4 2 5 5 6 9 ... n ... -7- 2 3 2 4 4 20 5 '3 10 14 7•• ~.~.~ , In this lesson you have learned some new techniques for fmding formulas for patterns involving division. In all the exercises we have restricted the rules to those that can be factored because most of the patterns you will encounter while making geometric discoveries in later chapters will be of this type. There are other techniques that can be used to find formulas for sequences that follow other rules. Inductive Reasoning I 55