Sequences A sequence is a function that computes an ordered list. For example, the average person in the United States uses 100 gallons of water each day. The function defined by (n) = 100n generates the terms of the sequence 100, 200, 300, 400, 500, 600, 700,…, when n = 1, 2, 3, 4, 5, 6, 7, …. This function represents the gallons of water used by the average person after n days.

Sequences A second example of a sequence involves investing money. If $100 is deposited into a savings account paying 5% interest compounded annually, then the function defined by g(n) = 100(1.05)n calculates the account balance after n years. The terms of the sequence are g(1), g(2), g(3), g(4), g(5), g(6), g(7), …, and can be approximated as 105, 110.25, 115.76, 121.55, 127.63, 134.01, 140.71, . . . .

Sequence A finite sequence is a function that has a set of natural numbers of the form {1, 2, 3, …, n} as its domain. An infinite sequence has the set of natural numbers as its domain.

Sequences For example, the sequence of natural-number multiples of 2, 2, 4, 6, 8, 10, 12, 14, …, is infinite, but the sequence of days in June, 1, 2, 3, 4, …, 29, 30, is finite.

Sequences Instead of using f (x) notation to indicate a sequence, it is customary to use an, where an = (n). The letter n is used instead of x as a reminder that n represents a natural number. The elements in the range of a sequence, called the terms of the sequence, are a1, a2, a3, …. The elements of both the domain and the range of a sequence are ordered. The first term is found by letting n = 1, the second term is found by letting n = 2, and so on. The general term, or nth term, of the sequence is an.

Sequences These figures show graphs of (x) = 2x and an = 2n. Notice that (x) is a continuous function, and an is discontinuous. To graph an, we plot points of the form (n, 2n) for n = 1, 2, 3,….

a. Solution Example 1 FINDING TERMS OF SEQUENCE Write the first five terms for each sequence. a. Solution Replacing n in with 1, 2, 3, 4, and 5 gives

b. Solution Example 1 FINDING TERMS OF SEQUENCE Write the first five terms for each sequence. b. Solution Replace n in with 1, 2, 3, 4, and 5 to obtain

Converge and Diverge If the terms of an infinite sequence get closer and closer to some real number, the sequence is said to be convergent and to converge to that real number. For example, the sequence defined by approaches 0 as n becomes large.

Converge and Diverge Thus an, is a convergent sequence that converges to 0. A graph of this sequence for n = 1, 2, 3, …, 10 is shown here. The terms of an approach the horizontal axis.

Converge and Diverge A sequence that does not converge to any number is divergent. The terms of the sequence are 1, 4, 9, 16, 25, 36, 49, 64, 81, …. This sequence is divergent because as n becomes large, the values of do not approach a fixed number; rather, they increase without bound.

b. Solution Example 2 USING A RECURSIVE FORMULA Find the first four terms of each sequence. b. Solution This is a recursive definition. We know a1 = 2 and an = an – 1 + n – 1.

MODELING INSECT POPULATION GROWTH Example 3 Frequently the population of a particular insect does not continue to grow indefinitely. Instead, its population grows rapidly at first, and then levels off because of competition for limited resources. In one study, the behavior of the winter moth was modeled with a sequence similar to the following, where an represents the population density in thousands per acre during year n.

a. Give the table of values for n = 1, 2, 3, …, 10 MODELING INSECT POPULATION GROWTH Example 3 a. Give the table of values for n = 1, 2, 3, …, 10 Solution Evaluate a1, a2, a3, …, a10. and

a. Give the table of values for n = 1, 2, 3, …, 10 MODELING INSECT POPULATION GROWTH Example 3 a. Give the table of values for n = 1, 2, 3, …, 10 Solution Approximate values for n = 1, 2, 3, …, 10 are shown in the table. n 1 2 3 4 5 6 7 8 9 10 an 2.66 6.24 10.4 9.11 10.2 9.31 10.1 9.43 9.98

MODELING INSECT POPULATION GROWTH Example 3 b. Graph the sequence. Describe what happens to the population density. Solution The graph of a sequence is a set of discrete points. Plot the points (1, 1), (2, 2.66), (3, 6.24), …,(10, 9.98), as shown here.

Series and Summation Notation Any sequence can be used to define a series. For example, the infinite sequence defines the terms of the infinite series

Series and Summation Notation If a sequence has terms a1, a2, a3, …, then Sn is defined as the sum of the first n terms. That is, The sum of the terms of a sequence, called a series, is written using summation notation. The symbol  , the Greek capital letter sigma, is used to indicate a sum.

Series A finite series is an expression of the form and an infinite series is an expression of the form The letter i is called the index of summation.

Caution Do not confuse this use of i with the use of i to represent the imaginary unit. Other letters, such as k and j, may be used for the index of summation.

Solution Example 4 USING SUMMATION NOTATION Evaluate the series Write each of the six terms, then evaluate the sum.

Solution Example 4 USING SUMMATION NOTATION Evaluate the series Write each of the six terms, then evaluate the sum.

Summation Properties If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n, (a) (b) (c) (d)