Arithmetic sequence
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In an arithmetic sequence, each term is the previous term summed with a constant. The constant we add to each term to get the next term is referred to as the common difference.
General form
The general form of an arithmetic sequence can be expressed as follows:
where:
is the first term is the common difference between the terms
Example
Consider the following arithmetic sequence:
Here we start with 2 and add 4 to get the next term in the arithmetic sequence.
Therefore in terms of the general form of the arithmetic sequence we can say:
Sum of arithmetic series
To compute the sum
We can use the following formula:
Example
Question. To calculate
Solution. Comparing to the general form we can see that
Derivation
We can write the equation for sum of an arithmetic series in the following two ways:
Adding the two equations together:
Note that there are
The key is that the red component can be re-written as: