Definition: A sequence is a function from a subset of the integers (usually either the set

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1 Math 3336 Section 2.4 Sequences and Summations Sequences Geometric Progression Arithmetic Progression Recurrence Relation Fibonacci Sequence Summations Definition: A sequence is a function from a subset of the integers (usually either the set {0, 1, 2, 3, 4, } or {1, 2, 3, 4, } ) to a set SS. We use the notation aa nn to denote the image of the integer nn. We call aa nn a term of the sequence. Example: Consider the sequence {aa nn }, where aa nn = 1. List the terms of this sequence beginning nn with aa 1. Geometric Progression Definition: A geometric progression is a sequence of the form: aa, aaaa, aarr 2, aarr 3,, aarr nn, where the initial term aa and the common ratio rr are real numbers. Examples: Write geometric progression with the following parameters: a. aa = 1 aaaaaa rr = 1 b. aa = 2 aaaaaa rr = 5 c. aa = 6 aaaaaa rr = 1/3 Page 1 of 6

2 Arithmetic progression Definition: An arithmetic progression is a sequence of the form: aa, aa + dd, aa + 2dd, aa + 3dd,, aa + nnnn, where the initial term aa and the common difference dd are real numbers. Example: Write geometric progression with the following parameters: a. a = 1 and d = 4 b. aa = 7 aaaaaa dd = 3 c. a = 1 and d = 2 Recurrence Relations Definition: A recurrence relation for the sequence {aa nn } is an equation that expresses aa nn in terms of one or more of the previous terms of the sequence, namely, aa 0, aa 1,, aa nn 1, for all integers nn with nn nn 0, where nn 0 is a nonnegative integer. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. The initial conditions for a sequence specify the terms that precede the first term where the recurrence relation takes effect. Example: Let {aa nn } be a sequence that satisfies the recurrence relation aa nn = aa nn for nn = 1,2,3,4, and suppose that aa 0 = 2. What are the first three terms of the sequence? Page 2 of 6

3 Example: Let {aa nn } be a sequence that satisfies the recurrence relation aa nn = aa nn 1 aa nn 2 for nn = 2,3,4, and suppose that aa 0 = 3 and aa 1 = 5. What are aa 2 and aa 3? Fibonacci sequence Definition: Define the Fibonacci sequence, ff 0, ff 1, ff 2, ff 3, by: Initial Conditions: ff 0 = 0, ff 1 = 1 Recurrence Relation: ff nn = ff nn 1 + ff nn 2 Example: Write the first six terms of the Fibonacci sequence. Solving recurrence relation Finding a formula for the nth term of the sequence generated by a recurrence relation is called solving the recurrence relation. Such a formula is called a closed formula. Various methods for solving recurrence relations will be covered in Chapter 8 where recurrence relations will be studied in greater depth. Here we illustrate by example the method of iteration in which we need to guess the formula. The guess can be proved correct by the method of induction (Chapter 5). Example (Iterative solution): Solve the recurrence relation and initial condition: aa nn = aa nn for nn = 2,3,4, and aa 1 = 2. Page 3 of 6

4 Example (Financial application): Suppose a person deposits $10,000 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years? More on solving recurrence relations Given a few terms of a sequence, try to identify the sequence. Conjecture a formula, recurrence relation, or some other rule. Some questions to ask? Are there repeated terms of the same value? Can you obtain a term from the previous term by adding an amount or multiplying by an amount? Can you obtain a term by combining the previous terms in some way? Are they cycles among the terms? Do the terms match those of a well-known sequence? Example: Find formulae for the sequences with the following first five terms: a. 1, 1 2, 1 4, 1 8, 1 16 b. 1, 3, 5, 7, 9 c. 1, -1, 1, -1, 1 Page 4 of 6

5 Summations Consider sum of the terms aa mm, aa mm+1,, aa nn of the sequence {aa nn }. We use notation nn jj=mm aa jj, mm jj nn aa jj to represent aa mm + aa mm aa nn. jj is called the index of summation, mm is the lower limit, and nn is the upper limit. Examples: a. jj b. 1 1 kk Example: Page 5 of 6

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