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  
th term. If  
  is the number that is added each time (called the common difference) and  
  is the first term, then the  
th term is  
For the sequence above,  
\[a=3, \: d=4\]
\[a_n=3+(n-1) \times 4=4n-1\]

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

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

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


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

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