If the sum of terms of an arithmetic sequence is identified by , find .
To find the thirteenth term in the sequence, when we know a formula for the sum of the first terms , we can simply find the sum of the first 13 terms and subtract the sum of the first 12 terms as follows:
Alternatively, we can find the formula for the term. We know for a arithmetic sequence the term is given by
where is the first term and is the common difference. We also know that the sum of a arithmetic sequence is given by
Comparing this with the equation we have been given, we can solve for and as follows:
By equating coefficients, we can see that , and (i.e., ). Hence, the equation for the term is
Using this formula, we can calculate .