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Introduction to arithmetic and geometric sequences.

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- 1. Sequences all around us patterns warped and otherwise by ﬂickr user Grant MacDonald
- 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. RANK 4, 7, 10, 13, 16 , 19 , 22
- 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. 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. 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. 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. 3, 6, 12, 24, , ,
- 9. 3, 6, 12, 24, , ,
- 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. 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. 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. 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. Write the implicit deﬁnition for this sequence. 32, 16, 8, 4, , ,
- 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:

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