Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
Welcome to our Math lesson on Series versus Sequences, this is the first lesson of our suite of math lessons covering the topic of Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series., you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
In the previous tutorial, we explained the concept of sequences, which are lists of numbers or other items that have a certain regularity governing the relationship between their terms. We dealt with arithmetic and geometric sequences (otherwise known as arithmetic and geometric progression), Fibonacci sequences, quadratic ones, etc. All of them have a first term x1 and a general term xn (expressed in terms of x1) and the rule that makes possible the calculation of any term.
In this tutorial we focus on the sum of a certain number of terms in a sequence. Such a sum is called a series. In correspondence to the sequence in question, we have arithmetic, geometric, Fibonacci-type, quadratic series etc. Although in mathematics the difference between the terms 'sequence' and 'series' is a bit blurred so that many people believe they represent the same thing, there is a key difference between them, as stated earlier. Sequences simply represent the list of all elements (terms) that are combined with each other through a certain rule, while a series represents the total or partial sum of the terms of the corresponding sequence. We represent a series with the letter Sn, where n is the number of the terms involved. For example, the arithmetic sequence 3, 6, 9, 12,…, has in correspondence the arithmetic series Sn = 3 + 6 + 9 + 12 + …., etc.
Another important difference between sequences and series consists of the fact that in sequences the order of terms matters. This means the two sequences 1, 3, 5 and 5, 3, 1 are not the same, as the first sequence is increasing (the common difference is d = 2) and the second is decreasing (d = -2). On the other hand, the two corresponding series are equal, as they give the same result (1 + 3 + 5 = 5 + 3 + 1 = 9).
Find the value of a, b and c in the arithmetic sequence below if S5 = 75.
Given that we are dealing with an arithmetic series, we have
Let's find the common difference d first. Thus, since the sum of the above terms is S5 = 75, we obtain
Thus,
In this way, we have
Enjoy the "Series versus Sequences" math lesson? People who liked the "Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. lesson found the following resources useful:
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Math tutorial "Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series." useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.