Many times a problem can be solved by reducing it to a problem that we already know how to solve.

Example 1

You are transported back in time to a classroom with a young Gauss. An exasperated instructor has asked the class to solve the following problem.

Problem: Compute the sum of the first \(100\) even integers:

\[2 + 4 + 6 + 8 + \dots + 200.\]

To do this, we can realize

\[2 + 4 + 6 + 8 + \dots + 200 = 2(1 + 2 + 3 + 4 + \dots + 100).\]

Since Gauss already figured out the sum of the first \(100\) integers is \(5050\) (see here), we automatically know the answer is \(10100\).

Example 2

What if instead Gauss’s teacher asked you to find the sum of the first \(100\) odd integers?

Problem: Calculate the sum of the first \(100\) odd integers

\[1 + 3 + 5 + 7 + \dots + 199.\]

To do this, we can realize that the sum of the first \(100\) integers is the same as the sum of the first \(200\) integers, minus the sum of the first \(100\) even integers:

\[1 + 3 + 5 + 7 + \dots + 199 = (1 + 2 + 3 + 4 + \dots + 200) - (2 + 4 + 6 + 8 + \dots + 200)\]

We know the first sum on the left hand side is \((200\cdot201)/2 = 20100\) (see here). Moreover, we figuredout that the second sum on the right hand side is \(10100\) in the previous example. So the sum we want is

\[1 + 3 + 5 + 7 + \dots + 199 = 20100 - 10100 = 10000.\]