**Linear Algebra**

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**Series and Sequences**

# Arithmetic sequence

*schedule*Aug 10, 2023

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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:

$a$ is the first term

$d$ 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:

$a$ = 2 (first term)

$d$ = 4 (common difference)

# Sum of arithmetic series

To compute the sum $S_n$ of an arithmetic series:

We can use the following formula:

## Example

Question. To calculate $S_n$:

Solution. Comparing to the general form we can see that $a=2$, $d=4$ and $n=6$. Now plugging these numbers into the formula for sum of arithmetic series:

## 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 $n$ terms altogether so the above simplifies to:

The key is that the red component can be re-written as: