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Chapter 12: Problem 45

Explain the basic difference between a sequence and a series.

Short Answer

Expert verified
A sequence is a list of numbers, while a series is the sum of those numbers.

Step by step solution

01

Define a Sequence

A sequence is an ordered list of numbers written in a definite order. Each number in the sequence is called a term. For example, 2, 4, 6, 8 is a sequence where each number follows a specific pattern (adding 2).
02

Understand a Series

A series is the sum of the terms of a sequence. When the terms of a sequence are added together, the result is a series. For example, for the sequence 2, 4, 6, 8, the corresponding series is 2 + 4 + 6 + 8.
03

Compare Sequence and Series

The fundamental difference between a sequence and a series is that a sequence is a list of numbers, while a series is the sum of that list of numbers. In other words, a sequence enumerates the individual elements, while a series aggregates them.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Definition
A **sequence** is an ordered list of numbers. Each number in this list is called a **term**. Sequences are often defined using a specific rule or formula that determines the relationship between the terms. For example, in the sequence 2, 4, 6, 8, each term increases by 2. Simple sequences can be defined by their pattern, such as:
  • **Arithmetic Sequences:** where each term is obtained by adding a constant value to the previous term. Example: 3, 6, 9, 12 (adding 3 each time).
  • **Geometric Sequences:** where each term is obtained by multiplying the previous term by a constant value. Example: 2, 4, 8, 16 (multiplying by 2 each time).

Sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely). Understanding the pattern or formula behind a sequence is crucial for analyzing its behavior.
Series Definition
A **series** is the sum of the terms of a sequence. When you add up all the terms in a sequence, the result is called a series. For example, for the sequence 2, 4, 6, 8, the corresponding series is 2 + 4 + 6 + 8.

Similar to sequences, there are different types of series:
  • **Arithmetic Series:** where the sum is taken over an arithmetic sequence. Example: 3 + 6 + 9 + 12.
  • **Geometric Series:** where the sum is taken over a geometric sequence. Example: 2 + 4 + 8 + 16.

The concept of series is important in various fields of mathematics and science because it helps in understanding the cumulative effect of adding terms in a sequence, whether they are finite or infinite. Formulas and methods, like the sum of an arithmetic series or the sum of a geometric series, can help in calculating the total value efficiently.
Sequence vs Series
Although sequences and series are closely related, they are fundamentally different concepts. Here are the key points to differentiate them:
  • **Sequence**: It is an ordered list of numbers. Each term stands alone and has its own place in the order. For Example, 2, 4, 6, 8.
  • **Series**: It is the sum of the terms of a sequence. Numbers are added together, and the focus is on their total sum. For Example, 2 + 4 + 6 + 8.

The relationship can be summarized as follows: A sequence lists the individual elements, whereas a series aggregates those elements into a single sum. Remember: If you see numbers listed out individually, that's a sequence. If you see those numbers being added up, you're looking at a series.

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Most popular questions from this chapter

Find a general term \(a_{n}\) for the given terms of each sequence. $$ \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots $$

Find the sum, if it exists, of the terms of each infinite geometric sequence. $$ \sum_{i=1}^{\infty} \frac{9}{8}\left(-\frac{2}{3}\right)^{i} $$

A particular substance decays in such a way that it loses half its weight each day. In how many days will 256 g of the substance be reduced to 32 g? How much of the substance is left after 10 days?

Write the first five terms of each geometric sequence. $$ a_{1}=5, r=-\frac{1}{5} $$

Evaluate the indicated term for each arithmetic sequence. $$ 2,4,6, \ldots ; \quad a_{24} $$
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