Identify the nature of the sequence. If the ratio of the second term to the first term equals the ratio of the third term to the second term, then it is geometric. If the difference between the second term and the first term equals the difference between the third term and the second term, then it is arithmetic.
Identify the final term of the series. If it is a given, like in the case of calculating the sum of the odd numbers from 1 to 100, then it is a finite series. If the sequence extends to an infinite number of terms, then it is an infinite series.
Calculate the ratio between terms in the case that the series is geometric. Divide any term by the previous term, and label this ratio "r."
Divide the first term by the difference between 1 and the ratio "r." This is the sum of an infinite series. For example, the sum of 1 + 1/2 + 1/4 + 1/8 + ... would be calculated as 1(1-1/2)/(1-1/2) = 1.
Multiply the first term by the difference between 1 and the exponent of "r" to the number of terms. Divide this by the difference between 1 and "r." This is the sum of a finite series.
Add the first and last terms of an arithmetic sequence. Multiply this sum by 1/2 the number of terms in the sequence. This is the sum of an arithmetic series. For example, to find the sum of all even numbers between 1 and 10, we know there are five terms; the first and last terms are 2 and 10, respectively, so we calculate the sum as (1/2)(5)(2+10) = 30.