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# Pre-Cal 40S June 3, 2009

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Introduction to arithmetic and geometric sequences.

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### Pre-Cal 40S June 3, 2009

1. 1. Sequences all around us patterns warped and otherwise by ﬂickr user Grant MacDonald
2. 2. Find the next three terms in each sequence of numbers ... 4, 7, 10, 13, , , 3, 6, 12, 24, , , 32, 16, 8, 4, , , 1, 1, 2, 3, 5, 8,13, , ,
3. 3. RANK 4, 7, 10, 13, 16 , 19 , 22
4. 4. 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. Example:
5. 5. Sequence: An ordered list of numbers that follow a certain pattern (or rule). 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. Example: 4, 7, 10, 13, , ,
6. 6. To Find The Common Difference d is the common difference tn is an arbitrary term in the sequence d = tn - t(n - 1) t(n - 1) is the term immediately before tn in the sequence Example: Find the common difference for the sequence: 11, 5, -1, -7, ... 5 - 11= -6 -1 - 5 = -6 d = -6 -7 - (-1) = -6
7. 7. To Find the nth Term In an Arithmetic Sequence t is the nth term n t = a + (n - 1)d a is the ﬁrst term n 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, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
8. 8. 3, 6, 12, 24, , ,
9. 9. 3, 6, 12, 24, , ,
10. 10. Geometic Sequence: (i) Recursive Deﬁnition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given ﬁrst term. (ii) Implicit Deﬁnition: An ordered list of numbers where each number in the list is generated by an exponential equation.
11. 11. Common Ratio (r): (i) The number that is repeatedly multiplied to successive terms in a geometic sequence. (ii) From the implicit deﬁnition, r is the base of the exponential function.
12. 12. To Find The Common Ratio r is the common ratio tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence
13. 13. To Find the nth Term In a Geometic Sequence tn is the nth term a is the ﬁrst term n is the quot;rankquot; of the nth term in the sequence r is the common ratio
14. 14. Write the implicit deﬁnition for this sequence. 32, 16, 8, 4, , ,
15. 15. Some quot;quickiesquot; to get us started ... Find the value(s) of r in . In the geometric sequence, if = 3 and r = 2 , ﬁnd . If the ﬁrst term of a geometric progression is and the common ratio is -3, ﬁnd the next three terms. Determine the common ratio for the geometric sequence: