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# Applied 20S January 7, 2009

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Arithmetic sequences and applications.

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### Applied 20S January 7, 2009

1. 1. Next in sequence? by ﬂickr user the mad LOLscientist
2. 2. Some Deﬁnitions Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence: (i) Recursive Deﬁnition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given ﬁrst term. (ii) Implicit Deﬁnition: An ordered list of numbers where each number in the list is generated by a linear equation. Common Difference (d): (i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit deﬁnition, d is the slope of the linear equation.
3. 3. To Find The Common Difference d is the common difference d = tn - t(n - 1) tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence To Find the nth Term In an Arithmetic Sequence tn is the nth term tn = a + (n - 1)d a is the ﬁrst term n is the quot;rankquot; of the nth term in the sequence d is the common difference Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ... Implicitly Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
4. 4. Which of the following sequences are arithmetic sequences? a) 1, 2, 6, 24, 120,… HOMEWORK b) 3, 9, 15, … c) 2, 4, 8, 16, 32,… d) 1, 2, 3, 5, 8, 13, … e) -4, -1, 2, 5, 8,…
5. 5. What is the pattern in the sequence 2, 8, 14, 20, 26…? Suggest an equation that could be used to generate such a list. HOMEWORK 2, 8, 14, 20, 26
6. 6. a) Why do the numbers 5, 8, 11, 14, 17… form an arithmetic sequence? HOMEWORK b) What is the deﬁning equation that produced them? c) What is the 27th term of this sequence? 5, 8, 11, 14, 17
7. 7. a) What is the next term of the sequence 1, -1, -3, -5, -7,…? b) Find an equation that could be used to generate such a sequence. c) What is the 35th term of this sequence?
8. 8. Harriet wants to phone her aunt in England to give her some family news. The ﬁrst minute of the phone call costs \$3.60 and each additional minute costs \$0.12. Set up an equation that could be used to deﬁne this relationship. Use the equation to calculate how much it would cost to talk for 28 minutes.
9. 9. List the ﬁrst 4 terms of the sequence determined by each of the following implicit deﬁnitions. (a) (b) (c) (d)
10. 10. Determine which of the following sequences are arithmetic. If a sequence is arithmetic, write the values of a and d. (a) 5, 9, 13, 17, ... (b) 1, 6, 10, 15, 19, ... (c)-1, -4, -7, -10, ... (d) x, 2x, 3x, 4x, ...
11. 11. Given the values of a and d, write the ﬁrst 5 terms of each arithmetic sequence. (a) a = 7, d, = 2 (b) a = -4, d, = 6 (c) a = 8, d, = x (d) a = 3m, d, = 1 - m
12. 12. Find the indicated terms for each arithmetic sequence. (a) 6, 8, 10, ... t10 and t4 (b) 9, 16, 23, ... t18 and t41 (c) -4, -9, -14, ... t18 and t66 (d) x, x + 4, x + 8, ... t 14 and t n
13. 13. Find the number of terms in each of the following arithmetic sequences. (a) 10, 15, 20, ..., 250 (b) 40, 38, 36, ...-30 (c) -2, -8, -14, ..., -206 (d) x + 2, x + 9, x + 16, ... , x + 303
14. 14. Complete each arithmetic sequence by ﬁnding the missing arithmetic means. (d) -1.5, ____, ____, ____, 4.5 (a) 1, ____, 25 (e) 2, ____, ____, ____, ____, 107 (b) 14, ____, ____, 32 (f) m + 40, ____, ____, ____, m + 4 (c) -3, ____, ____, -60
15. 15. The eighth term of an arithmetic sequence is 5.3 and the fourteenth term is 8.3. What is the ﬁfth term?
16. 16. Answers to the last 7 questions: 1. (a) 0, 3, 6, 9 (b) 0, 1, 4, 9 (c) 1, 2, 4, 8 (d) (c) -1, -3 (d) x, x 2. (a) 5, 4 (b) not arithmetic 3. (a) 7, 9, 11, 13, 15 (b) -4, 2, 8, 14, 20 (c) 8, 8 + x, 8 + 2x, 8 + 3x, 8 + 4x (d) 3m, 2m + 1, m + 2, 3, 4 - m (d) x + 4(n - 1), x + 52 4. (a) 24, 72 (b) 65, 702 (c) -89, -329 5. (a) 49 (b) 36 (c) 35 (d) 44 6. (a) 13 (b) 20, 26 (c) -22, -41 (f) m + 31, m + 22, m + 13 (d) 0, 1.5, 3 (e) 23, 44, 65, 86 7. 3.8