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Chapter 13: Problem 55

What is the difference between a sequence and a series?

Short Answer

Expert verified
A sequence is an ordered list of numbers; a series is the sum of the numbers in a sequence.

Step by step solution

01

- Define Sequence

A sequence is a list of numbers or terms arranged in a specific order. Each number in the sequence is called a term. For example, the sequence 2, 4, 6, 8, 10, ... has its terms listed in a specific progression.
02

- Define Series

A series is the sum of the terms of a sequence. If we take the sequence 2, 4, 6, 8, 10, ..., the series formed by summing these terms would be 2 + 4 + 6 + 8 + 10 + ....
03

- Highlight the Difference

The main difference is that a sequence is simply an ordered list of numbers, while a series is the sum of the numbers in a sequence.

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Key Concepts

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

Sequence
A sequence is essentially a list of numbers arranged in a particular order. Each number in this list is called a *term*. For example, consider the sequence: 2, 4, 6, 8, 10, .... Here, 2 is the first term, 4 is the second term, 6 is the third, and so on. Sequences can follow various patterns and rules, such as arithmetic progression (adding a constant value each time) or geometric progression (multiplying by a constant value each time).
Series
While a sequence is a list of numbers, a series is what you get when you sum the terms of that sequence. Using our earlier sequence example: 2, 4, 6, 8, 10, ..., if you add these terms together, you get the series: 2 + 4 + 6 + 8 + 10 + .... Series can be either finite (with a limited number of terms) or infinite (continuing indefinitely). Essentially, a series gives us the sum based on the terms of a sequence.
Terms in a Sequence
Each number in a sequence is referred to as a *term*. For instance, in the sequence 3, 6, 9, 12, 15, ..., 3 is the first term, 6 is the second term, 9 is the third term, and so on. Knowing the position of each term helps in identifying and understanding the pattern or rule that the sequence follows. It's essential to recognize each term to effectively work with sequences and series.
Sum of Sequence Terms
The sum of the terms in a sequence leads to the formation of a series. To find this sum, you add up the individual terms of the sequence. For example, let's take the sequence: 1, 3, 5, 7, 9. The series would be 1 + 3 + 5 + 7 + 9. Calculating this sum involves simple addition but becomes more complex with longer sequences or specific patterns. In some cases, there are formulas to find sums for specific types of sequences, like arithmetic or geometric sequences.

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Most popular questions from this chapter

Solve each problem using the ideas of arithmetic sequences and series. On the first day of October an English teacher suggests to his students that they read five pages of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during October?

Write the first four terms of the infinite sequence whose nth term is given. \(c_{n}=(-1)^{n}(2 n-1)^{2}\)

Discussion. Find the fourth term in the binomial expansion for \((x+y)^{120} .\) Find the fifth term in the binomial expansion for \((x-2 y)^{100} .\) Did you have any trouble computing the coefficients?

Discussion. Find the trinomial expansion for \((a+b+c)^{3}\) by using \(x=a\) and \(y=b+c\) in the binomial theorem.

Find the sum of each series. $$\sum_{j=0}^{3}(j+1)^{2}$$
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