Sum of even numbers N?

Question

To sum first N numbers we can use this formula:

$$1 + 2 + 3 + \ldots + n = \frac{(n (n + 1)}{2}$$

To sum even numbers we multiply this formula by 2:

$$2 + 4 + 6 + \ldots + 2n = n (n + 1)$$

Lets check sum of even numbers until 6:

Sum(odd) =  2 + 4 + 6 = 12

Then let's use the formula: 6 * (6 + 1) = 6 * 6 + 6 = 42.

What is wrong with formula or my calculations?

Answer 1

$2+4+6=12$ is the sum of the first three even numbers, and indeed for $n=3$ you have that $3\cdot (3+1)=12$

If you set $n=6$, you are calculating the sum of the first six even numbers, and indeed $2+4+6+8+10+12=42$

So, there is nothing wrong with your formula: it is correct. You just misapplied it.

Answer 2

You use that

$$2+4 + 6+ \ldots + 2n = 2(1 + 2 + 3 + \ldots + n) = 2\cdot \frac{n(n+1)}{2} = n(n+1)$$

So the $n$ used is the number of terms added, not the final term, but half of it.