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SequencesSection 8.1

Sequences, Series,and Probability

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8.1

Definition: Sequence

A finite sequence is a function whose domain is {1, 2, 3, … , n}, the positive integers less than or equal to a fixed positive integer n.

An infinite sequence is a function whose domain is the set of all positive integers.

Definition

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The function f(n) = n2 with domain {1, 2, 3, 4, 5} is a finite sequence.

For the independent variable of a sequence, we usually use n (for natural number) rather than x and assume that only natural numbers can be used in place of n.

For the dependent variable f(n), we generally write an (read “a sub n”).

So this finite sequence is also defined by

an = n2 for 1 ≤ n ≤ 5.

Definition

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The terms of the sequence are the values of the dependent variable an .

We call an the nth term or the general term of the sequence.

The equation an = n2 provides a formula for finding the nth term.

Definition

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8.1

Definition: Factorial Notation

For any positive integer n, the notation n! (read “n factorial”) is defined by

The symbol 0! is defined to be 1, 0! = 1.

.123 1! nnn

In general, n! is the product of the positive integers from 1 through n. The value given to 0! is 1.

Factorial Notation

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8.1

A recursion formula gives the nth term as a function of the previous term.

If the first term is known, then a recursion formula determines the remaining terms of the sequence.

Recursion Formulas

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8.1

An arithmetic sequence can be defined as a

sequence in which there is a common difference

d between consecutive terms, or it can be defined

by giving a general formula that will produce such

a sequence.

Arithmetic Sequences

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8.1

Definition: Arithmetic Sequence

A sequence that has an nth term of the form

an = a1 + (n – 1)d,

where a1 and d are any real numbers, is called an arithmetic sequence.

Note that an = a1 + (n – 1)d can be written as an = dn + (a1 – d). So the terms of an arithmetic sequence can also be

described as a multiple of the term number plus a constant, (a1 – d).

Arithmetic Sequences