Pre-Cal 40S Slides June 2, 2008
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This document defines sequences and provides information about arithmetic and geometric sequences. It discusses the recursive and implicit definitions of sequences. For arithmetic sequences, it explains how to find the common difference and the nth term using the common difference and the first term. For geometric sequences, it similarly explains how to find the common ratio and the nth term using the common ratio and first term.
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The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
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The document defines sequences and provides information about arithmetic and geometric sequences. It explains that:
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The document provides information about different types of infinite sequences, including explicitly defined sequences, recursively defined sequences, arithmetic sequences, harmonic sequences, and geometric sequences. It gives examples and formulas for finding terms, sums, and analyzing properties of each type of sequence. Key points covered include the definitions of explicit and recursive sequences, the formulas for the nth term and partial sum of arithmetic and geometric sequences, and examples of finding terms and analyzing sequences.
Number Sequences
Here are the nth terms for the given sequences:
(a) The nth term is: 3n + 1
(b) The nth term is: n + 2
(c) The nth term is: n + 1
(d) The nth term is: 10n
(e) The nth term is: 5n - 1
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Pre-Cal 40S Slides June 2, 2008
- 1. Sequences Superman Sequence by flickr user nib.ipernity.com
- 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. 4, 7, 10, 13, 16 , 19 , 22
- 4. 3, 6, 12, 24, , ,
- 5. Write the implicit definition for this sequence. 32, 16, 8, 4, , ,
- 6. 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.
- 7. 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, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) Implicitly d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
- 8. Geometric 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. 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.
- 9. 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 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