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12.1 Sequences and Summation Notation

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Objectives

► Sequences

► Recursively Defined Sequences

► The Partial Sums of a Sequence

► Sigma Notation

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Sequences and Summation Notation

Roughly speaking, a sequence is an infinite list of numbers. The numbers in the sequence are often written as a1, a2, a3, . . . . The dots mean that the list continues forever. A simple example is the sequence

We can describe the pattern of the sequence displayed above by the following formula:

an = 5n

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Sequences

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SequencesAny ordered list of numbers can be viewed as a function whose input values are 1, 2, 3, . . . and whose output values are the numbers in the list. So we define a sequence as follows:

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Example 1 – Finding the Terms of a Sequence

Find the first five terms and the 100th term of the sequence defined by each formula.

(a) an = 2n – 1

(b) cn = n2 – 1

(c)

(d)

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Example 1 – SolutionTo find the first five terms, we substitute n = 1, 2, 3, 4, and 5 in the formula for the nth term.

To find the 100th term, we substitute n = 100. This gives the following.

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Example 2 – Finding the nth Term of a Sequence

Find the nth term of a sequence whose first several terms are given.

(a)

(b) –2, 4, –8, 16, –32, . . .

Solution:(a) We notice that the numerators of these fractions are the

odd numbers and the denominators are the evennumbers. Even numbers are of the form 2n, and oddnumbers are of the form 2n – 1 (an odd number differsfrom an even number by 1).

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Example 2 – SolutionSo a sequence that has these numbers for its first fourterms is given by

(b) These numbers are powers of 2, and they alternate in sign, so a sequence that agrees with these terms isgiven by

an = (–1)n2n

You should check that these formulas do indeedgenerate the given terms.

cont’d

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Recursively Defined Sequences

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Recursively Defined SequencesSome sequences do not have simple defining formulas like those of the preceding example.

The nth term of a sequence may depend on some or all of the terms preceding it.

A sequence defined in this way is called recursive. Here is an example.

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Example 4 – The Fibonacci Sequence

Find the first 11 terms of the sequence defined recursively by F1 = 1, F2 = 1 and Fn = Fn –1 + Fn –2

Solution:To find Fn, we need to find the two preceding terms, Fn –1and Fn –2. Since we are given F1 and F2, we proceed as follows.

F3 = F2 + F1 = 1 + 1 = 2

F4 = F3 + F2 = 2 + 1 = 3

F5 = F4 + F3 = 3 + 2 = 5

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Example 4 – SolutionIt’s clear what is happening here. Each term is simply the sum of the two terms that precede it, so we can easily write down as many terms as we please.

Here are the first 11 terms:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

cont’d

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Recursively Defined SequencesThe sequence in Example 4 is called the Fibonacci sequence, named after the 13th century Italian mathematician who used it to solve a problem about the breeding of rabbits.

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The Partial Sums of a Sequence

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The Partial Sums of a SequenceIn calculus we are often interested in adding the terms of a sequence. This leads to the following definition.

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Example 5 – Finding the Partial Sums of a Sequence

Find the first four partial sums and the nth partial sum of the sequence given by an = 1/2n.

Solution:The terms of the sequence are

The first four partial sums are

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Example 5 – Solution

Notice that in the value of each partial sum, the denominator is a power of 2 and the numerator is one less than the denominator.

cont’d

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Example 5 – SolutionIn general, the nth partial sum is

The first five terms of an and Sn are graphed in Figure 7.

Graph of the sequence an and the sequence of partial sums SnFigure 7

cont’d

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Sigma Notation

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Sigma NotationGiven a sequence

a1, a2, a3, a4, . . .

we can write the sum of the first n terms using summation notation, or sigma notation. This notation derives its name from the Greek letter (capital sigma, corresponding to our S for “sum”). Sigma notation is used as follows:

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Sigma NotationThe left side of this expression is read, “The sum of ak from k = 1 to k = n.”

The letter k is called the index of summation, or the summation variable, and the idea is to replace k in the expression after the sigma by the integers 1, 2, 3, . . . , n, and add the resulting expressions, arriving at the right side of the equation.

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Example 7 – Sigma NotationFind each sum.

Solution:

= 12 + 22 + 32 + 42 + 52

= 55

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Example 7 – Solution

= 5 + 6 + 7 + 8 + 9 + 10

= 45

= 2 + 2 + 2 + 2 + 2 + 2

= 12

cont’d

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Sigma NotationThe following properties of sums are natural consequences of properties of the real numbers.