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In addition to the revision notes for Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. on this page, you can also access the following Sequences and Series learning resources for Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series.
In these revision notes for Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series., we cover the following key points:
By definition, a number series is the sum of the first n terms of a number sequence. In correspondence to the sequence in question, we have arithmetic, geometric, Fibonacci-type, quadratic series etc.
The main difference between sequences and series consists of the fact that while sequences represent simply the list of all elements (terms) that are combined with each other through a certain rule, a series instead 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.
Another important difference between sequences and series consists of the fact that in sequences the order of terms matters. This is because the common difference or ratio changes when we change the order of a sequence but the sum of terms calculated when dealing with the corresponding series is the same.
By definition, an arithmetic series Sn represents the sum of the first n terms of an arithmetic sequence (progression). This means the main objective when dealing with an arithmetic series is to find the sum of the first n terms of the corresponding arithmetic progression (sequence). For this, we often use the Gauss formula
where x1 is the first term and xn is the nth term of the given arithmetic series.
We can also use another formula for the calculation of the first n terms of an arithmetic series - a formula obtained by expressing xn in the Gauss formula in terms of the first term x1 and the common difference d. It is
By definition, a geometric series Sn represents the sum of the first n terms of a geometric sequence (progression).
The general formula for calculating the sum of the first n terms of a geometric series (for the common ratio R > 1) is
Multiplying the above formula up and down by -1 gives another version of the sum of the first n terms of a geometric progression (series), i.e.
This version is used in decreasing geometric sequences, where the common ratio R is smaller than 1.
In some questions, there are certain situations where there is missing information, which is completed by combining the sequences and series approach.
Knowing how to deal with arithmetic and geometric series allows finding many more things related to combined series, such as for example in series with rational terms, where numerators form a different pattern from denominators.
Number series and sequences are very common in practice. We can find them everywhere, from banking to construction; from physics to engineering and so on.
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