An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.

Since this difference is common to all consecutive pairs of terms, it is called the common difference. It is denoted by d. If the difference in consecutive terms is not constant, then the sequence is not arithmetic. The common difference can be found by subtracting two consecutive terms of the sequence.

The formula for the common difference of an arithmetic sequence is: d = a_{n+1} -
a_{n}

An arithmetic sequence is a linear function. Instead of y=mx+b, we write a_{n}=dn+c
where d is the common difference and c is a constant (not the first term of
the sequence, however).

A recursive definition, since each term is found by adding the common difference
to the previous term is a_{k+1}=a_{k}+d.

For any term in the sequence, we've added the difference one less time than the number of the term. For example, for the first term, we haven't added the difference at all (0 times). For the second term, we've added the difference once. For the third term, we've added the difference two times.

The formula for the general term of an arithmetic sequence is: a_{n} =
a_{1} + (n-1) d

A series is a sum of a sequence. We want to find the n^{th} partial
sum or the sum of the first n terms of the sequence. We will denote the n^{th} partial
sum as S_{n}.

Consider the arithmetic series S_{5} = 2 + 5 + 8 + 11 + 14. There
is an easy way to calculate the sum of an arithmetic series.

S_{5} = 2 + 5 + 8 + 11 + 14

The key is to switch the order of the terms. Addition is commutative, so changing the order doesn't change the sum.

S_{5} = 14 + 11 + 8 + 5 + 2

Now, add those two equations together.

2*S_{5} = (2+14) + (5+11) + (8+8) + (11+5) + (14+2)

Notice that each of those sums on the right hand side is 16. Instead of writing 16 (the sum of the first and last terms) five times, we can write it as 5 * 16 or 5 * (2 + 14)

2*S_{5} = 5*(2 + 14)

Finally, divide the whole thing by 2 to get the sum and not twice the sum

S_{5} = 5/2 * (2 + 14)

I've purposely not simplified the 2+14 so that you can see where the numbers come from. This sum would be 5/2 *(16) = 5(8) = 40.

Now, if we try to figure out where the different parts of that formula come
from, we can conjecture about a formula for the n^{th} partial sum.
The 5 is because there were five terms, n. The 16 is the sum of the first and
last terms, a_{1} + a_{n}. The 2 is because we added the sum
twice and will remain a 2. Therefore, the sum of the first n terms of an arithmetic
sequence is S_{n}=n/2*(a_{1}+a_{n})

There is another formula that is sometimes used for the n^{th} partial
sum of an arithmetic sequence. It is obtained by substituting the formula for
the general term into the above formula and simplifying. The preferred method
is to go ahead and find the n^{th} term, and then just plug that number
into the formula.

S_{n} = n/2 * ( 2a_{1} + (n-1) d )

Find the sum from k=3 to 17 of (3k-2).

The first term is found by substituting k=3 into 3k-2 to get 7. The last term is 3(17)-2 = 49. There are 17 - 3 + 1 = 15 terms. So, the sum is 15 / 2 * (7 + 49) = 15 / 2 * 56 = 420.

Note that there are 15 terms there. When the lower limit of the summation is 1, there is little problem figuring out what the number of terms is. However, when the lower limit is any other number, it seems to give people difficulty. No one would argue that if you went from 1 to 10, there are 10 numbers. However, the difference between 10 and 1 is only 9. So, when you are finding the number of terms, it is the upper limit minus the lower limit plus one.