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Chapter 5: Problem 135

What is a sequence? Give an example with your description.

Short Answer

Expert verified
A sequence is a function which maps natural numbers to a set of real numbers, represented by \( a_n \). In simpler terms, it is a set of numbers or objects in a specific order. An example of a sequence can be an arithmetic sequence such as \(2, 4, 6, 8, 10,...\).

Step by step solution

01

Definition of a Sequence

A sequence in mathematics is a function, which maps the set of natural numbers (mostly described as \( n \)) to a set of real numbers. The output of each function is defined as \( a_n \), which is the general term of the sequence.
02

Properties of a Sequence

Sequences can be finite or infinite. That means a sequence can have a limited or an unlimited number of terms. Also, the terms of a sequence could be numbers, variables, or expressions made from numbers and variables.
03

Representation of a Sequence

The sequence can be represented in three ways: either by List representation, Set representation or Rule representation. If it is finite, all the terms can be listed. In the case of an infinite sequence, we list as many terms as possible and then indicate the pattern continues indefinitely. In Set representation, we describe the terms of the sequence as a set. And in the Rule representation form, we provide a formula that generates the terms of the sequence.
04

Example of a Sequence

Here's an example of a sequence, which is an arithmetic sequence: \(2, 4, 6, 8, 10,...\). This sequence has a common difference of 2 between each consecutive term, and it is infinite as it goes on indefinitely. The general term which produces the sequence could be represented as \( a_n = 2n \) where \( n \) is each natural number.

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(7,-7,-21,-35, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{4}, r=\frac{1}{2}\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=0.1\)

Company A pays $$\$ 23,000$$ yearly with raises of $$\$ 1200$$ per year. Company B pays $$\$ 26,000$$ yearly with raises of $$\$ 800$$ per year. Which company will pay more in year 10 ? How much more?

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(1.5,-3,6,-12, \ldots\)
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