# Finding the sum of a sequence of integers

When a question asks you to add a large number of integers, it can seem as if the answer may take some time to find. But the GMAT is all about smart shortcuts. Here's a quick method that should make you happy to see such questions on test day.

Consider the following question from GMAT Tutor:

*For every positive integer n, the nth term of a sequence is the sum of three consecutive integers starting at n. What is the sum of terms 1 through 99 of this series?*

## Rules for finding the sum of this sequence of integers quickly

**Rule #1: The sum of a sequence of integers is the average of the sequence of integers multiplied by the number of terms.**

If this rule sounds familiar, it is because it is simply another version of the formula for finding an average:

- Average = Sum of terms / Number of terms

Now, how do we determine the average of a sequence of integers?

**Rule #2: The average of a sequence of integers is the average of the first and last terms**

## Applying the rules to find the sum of the sequence

How do we apply these useful rules to this question?

**First, calculate the average of the first and last terms.**

- The first term is the sum of 1, 2 and 3 = 6
- The last term is the sum of 99, 100 and 101 = 300
- The average of the first and last terms = (6 + 300) / 2 = 306 / 2 = 153

**Second, multiply the average by the number of terms.**

- There are 99 terms
- Therefore, the answer to our question is 153 x 99

Note that a quick way to calculate this without a calculator is to multiply 153 x 100 and subtract 153.

- 153 x 100 = 15,300
- 15,300 – 153 =
**15,147**

And that is our final answer!

You can see how the application of simple rules makes questions that seem difficult much easier. Your task is to assemble as many such rules as you can.

_{This was a sample of the in-depth instruction that GMAT Tutor offers about solving problems in the GMAT Quant section. For complete and interactive lessons and online tutor support, subscribe to one of GMAT Tutor's top-rated GMAT prep plans. Commitment-free trials are available for seven days.}