Efficient Equation for Counting Even Numbers from 1 to n

[Genericcoder]

Hi guys,

I am noobie in number theory so if something exists better than this equation I did please don't bash me this is my first equation.

Today I was thinking of a way to count all even from 1 to n,so this way I thought about is like this.

Let me first write the equation,then I will explain the logic.


iNumber = ((iNumber + 1) * (iNumber / 2)) - (((iNumber + 1) * (iNumber / 2) - (iNumber / 2)) / 2)


Okay let me explain how I got this logic,so let's pick odd numbers from 1 to 10 and see how they form.
1 = 0 + 1;
3 = 2 + 1;
5 = 4 + 1;
7 = 6 + 1;
9 = 8 + 1;

So here we have 5 ones,so for example if we removed those ones from the a equation that you add 1 to 10 from it will be like this;

Normal numbers which you added 1 to 10 ->

1 + 2 + 2 + 1 + 4 + 4 + 1 + 6 + 6 + 1 + 8 + 8 + 1 + 10 -
1 + 1 + 1 + 1 + 1,so the numbers now become

0 + 2 + 2 + 4 + 4 + 6 + 6 + 8 + 8 + 10 = 50,but that's 2x,so if we got that number and divide by 2 it will be 25,thats how much odds add up,so if we got the total numbers from 1 to 10 and minus 25 we should get the even number we want;

So let's put that in the equation and see if its right;

((11) * (5)) - (((11) * (5) - (5)) / 2) = 55 - 25 = 30;

2 + 4 + 6 + 8 + 10 = 30;

So the equation is right I am sure this has been done by another mathmetician,but its good to think about it the logic of it is great.

 

[dodo]

Hi, genericcoder,
I'll try to illustrate how your formula connects with other formulas known to mathematicians; in other words, how to prove (and simplify) your formula, if this is of any use to you.

A formula well known by mathematicians (and possibly by you) is a formula for the sum of the first consecutive integers; for example, 1 + 2 + 3 + 4 + 5. The formula is$$\frac {k (k+1)} 2$$In this example, 1 + 2 + 3 + 4 + 5 = 5 * 6 / 2 = 30 / 2 = 15.

You want to calculate the sum of consecutive even numbers, such as 2 + 4 + 6 + 8 + 10, and that is twice the sum 1 + 2 + 3 + 4 + 5. So, if your number is 10, you would apply the above formula with 10/2 = 5 (which gives you 1+2+3+4+5; we did that and obtained 5*6/2 = 15) and then multiply the result by two to obtain 2+4+6+8+10 (= 15*2 = 30).

So the sum of the first even numbers can be expressed as twice the above formula, that is, $k(k+1)$, but using a value of $\displaystyle {k=\frac n 2}$:$$\begin{align*}\frac n 2 \left( \frac n 2 + 1 \right) &= \frac {n^2} 4 + \frac n 2 = \frac{n^2 + 2n} 4 \\ &= \frac {n(n+2)} 4 & \mbox{(eq. 1)}\end{align*}$$
In your example, if you evaluate $\displaystyle {\frac {n(n+2)} 4}$ with $n=10$, you obtain 10*12/4 = 30, as you did.

The formula you give can be simplified to this one marked as (eq. 1), with a little algebra. Your formula is$$\displaystyle {(n+1)\frac n 2 - \frac {(n+1)\frac n 2 - \frac n 2} 2}$$
Notice that the numerator of the big fraction on the right is $\displaystyle (n+1)\frac n 2 - \frac n 2$; in other words, (n+1)*something - something. If you have (n+1) of something, and subtract one of something, you are left with n*something. So we begin by simplifying your formula to$$\displaystyle {(n+1)\frac n 2 - \frac {n\frac n 2} 2}$$
or$$(n+1)\frac n 2 - \frac {n^2} 4$$
Expanding the parentheses on the left part, we obtain$$\begin{align*}& \frac {n^2} 2 + \frac n 2 - \frac {n^2} 4 \\ &= \frac {n^2} 4 + \frac n 2 \\ &= \frac {n(n+2)} 4 \end{align*}$$
just as we did above for (eq. 1).

Hope this helps!

 

[Genericcoder]

@Dodo

Thanks a lot that makes sense,also this logic for calculating the even is more efficient than mine I made it more complex.