Just adding

According to legend, when the mathematician Carl Friedrich Gauss was 7 years old, his first grade teacher assigned the class to add up the first one hundred natural numbers.  The teacher of course hoped that this would occupy the students for a while and refused to believe young Gauss when the latter almost immediately claimed to have completed the task.  The answer was correct, and the quick solution consisted mainly of one simple observation: \[1+100 = 2+99 = 3+98 =\cdots = 50+51.\]  Thus, adding 1 through 100 is equal to adding 101 to itself 50 times so the answer is \(101*50=5050\).

Here is a geometric way of seeing what is going on (where we use 4 instead of 100 as our example).
The first row contains one pin, the second one contains two, the third three and the fourth four pins.

 Now if we rotate this configuration 180 degrees and add it to itself we get the following picture:

We can easily see that we have four rows with five pins each and so the total number of pins is
\(4 \times 5 = 20\).  Hence \(2 \times (1+2+3+4)=20\) and so \(1+2+3+4=10\).  Yes, we could have
easily computed this in our heads, but now we can see that we can replace 4 in this picture by any arbitrary integer $n$ and we will have $n$ rows with $n+1$ pins each, and so we obtain the formula
\[1+2+\cdots+n=\frac{n(n+1)}{2}.\]

This formula can also be proved using the principle of mathematical induction (which we will not do here as it is not particularly enlightening) and is a classic example for teaching someone about induction.

Now try to find a formula for the sum of the first $n$ odd numbers and one for the sum of the first $n$ even numbers both by using Gauss' trick and geometrically.