Problem 55 - Algebra for College Students

Question

What is the difference between a sequence and a series?

Step-by-Step Solution

Answer

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

Step 1 - 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.

Step 2 - 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 + ....

Step 3 - 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.

Key Concepts

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.