## Arithmetic Sequences

An arithmetic sequence is any sequence where a fixed number is added to a term to get the next term.
3, 7, 11, 15, 19
is an arithmetic sequence with 4 being added to each term to get the next term.
Given any arithmetic sequence we can find an expression for the
$n$
th term. If
$d$
is the number that is added each time (called the common difference) and
$a$
is the first term, then the
$n$
th term is
$a_n=a+(n-1)d$
.
For the sequence above,
$a=3, \: d=4$
.
Hence
$a_n=3+(n-1) \times 4=4n-1$

We can also find a formula for the sum
$S_n$
of the first
$n$
terms.
$S_n=a+(a+d)+...(a+(n-2)d)+ (a+(n-1)d)$

Writing this sum backwards gives
$S_n=(a+(n-1)d)+(a+(n-2)d)+...+ (a+d)+a$

Now adding these two sums gives
\begin{aligned} 2S_n &= \underbrace{(a+(n-1)d)+(a+(n-1)d)+...+ (a+(n-1)d)+(a+(n-1)d)}_{n \: terms} \\ &= n(2a+(n-1)d) \end{aligned}

Hence
$S_n=\frac{n}{2}(2a+(n-1)d)$

For the sequence above the sum of the first 20 terms is
$S_{20}=\frac{20}{2} \times 3+(20-1) \times 4)=820$