A series is a little bit different from a sequence: a sequence is nothing more than an ordered list of numbers; a series is that same list combined with the addition operation.

For example, {1, 2, 3, 4} is an arithmetic sequence, and the arithmetic series that goes with this sequence is:

1 + 2 + 3 + 4. We say that the sum of the series, then, is 10.

A story is told about the mathematician Carl Frederick Gauss, who as a student was told to add all the integers from 1 to 100. Without wasting any time, he simply wrote down the answer 5050. When his teacher demanded to know how he got the answer so quickly, he explained as follows:

If you group them in pairs, from opposite ends of the series, you get 1 + 100, 2 + 99, 3 + 98, and so on.

Each of those pairs adds to 101, and there are 50 such pairs, so the sum is 101 x 50 = 5050.

The reasoning is correct, and leads us to the following formula, in which s_n represents the sum of the first n terms:

s_n = ⁿ/₂(a₁ + a_n)

Of course, we could write a_n = a₁ + (n - 1)d, so that helps us out if we know the first term and the common difference.

I like putting formulas into words, because it helps me remember them, so I say that the sum of the series is the average of the first and last term, divided by the number of terms.

Example: If the first term is 5, and the common difference is 2, what is the sum of the first 10 terms?

Answer: a₁₀ = 5 + 9 x 2 = 23.

Averaging the first and tenth terms gives us (5 + 23)/2 = 14. Multiplying this by the number of terms (10) gives us a sum of 140.

Example: If the first term is 3, and the common difference is 4, what is the sum of the third through 10th terms?

Answer: The third term is 3 + 4 x 2 = 11, and the tenth term is 3 + 4 x 9 = 39. The average is (11 + 39)/2 = 25. However, since we are starting with the 3rd term, we have a total of 8 terms, so the total is 25 x 8 = 200.

Example: If the first term is 5, and the second term is 3, find the sum of the first 20 terms.

Answer: The common difference in this case is -2. So the 20th term is 5 - 19 x 2 = -33. The average of the first and 20th terms is -14. There are twenty terms, so the sum is -14 x 20 = -280.

1.

If the first term of an arithmetic series is 12, and the common difference is -1, what is the sum of the first 10 terms?

2.

If the first term of an arithmetic series is 10, and the third term is 3, find the sum of the first 10 terms.

3.

The third and fourth terms of an arithmetic series are 14 and 18. What is the sum of the first 15 terms?

4.

The first term of an arithmetic series is -2, and the second term is 2. What is the sum of the third through 8th terms?

5.

The sum of the first 5 terms of an arithmetic series is 50. The sum of the first 7 terms of the same series is 91. What is the first term of the series?

6.

If the first two terms of an arithmetic series are 0 and 1, what is the sum of the 10th through 20th terms?

7.

If the second term of an arithmetic series is twice the first term, and the third term is twice the second term, what is the sum of the first 10 terms?

8.

The tenth term of an arithmetic series is 12 more than the eighth term. If the second term is 5, what is the sum of the first 5 terms?

9.

The sum of the first nine terms of an arithmetic series is 72, and the first term is 6. What is the common difference?

10.

The sum of the first 20 terms of an arithmetic series is 2200, and the common difference is 12. What is the tenth term?