What is the name of this formula for finding the sum of consecutive numbers?

  • Thread starter RobinSky
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In summary: You don't need a "name" for every algebraic manipulation!In summary, the formula for the difference of two squares, which is (x+1)^2-x^2=2x+1, is also known as the "difference of two squares" theorem. It is a useful formula in algebra and can be applied to various numbers, as shown in the examples of 12^2 and 13^2 and 100^2 and 101^2. Additionally, it is closely related to the discoveries of Pythagoras in regards to the sum of all odd numbers being a square.
  • #1
RobinSky
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I found out that if you take for example 13^2-12^2 you get 13+12 and this works for all numbers, the formula is (x+1)^2-x^2=2x+1

I'm not sure if this formula is famous/known but I'm sure it is and my question is; is there a name for it and if so, what's the name for it?
 
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  • #2
The more general form a2 - b2 = (a + b)(a - b) is just called 'difference of two squares' and is fairly constantly useful as is the more general difference of two n powers which gives you the formula of the sum of a geometric series. In your quoted cases the difference happens to be 1, but I don't know that there is a special name for that (I do know one is not needed).
 
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  • #3
12^2 = 144
13^2 = 169

169 - 144 = 12 + 13 = 25

Then it just keeps on going, with every single numbers followed by each other.

100^2 = 10000
101^2 = 10201

10201 - 10000 = 100 + 101 = 201

Sure you knew that this was what I am talking about?:smile:
 
  • #4
Maybe this helps as well, not sure but also a formula I came up with.

x^2+x+n=n^2

Let's now say x is 12 again. n = x+1

12^2+12+13=13^2

144+24+1=169.

What I'm trying to ask is, what's the name of this "thingy"(formula/teorem what ever)? If you choose any number for the variabel x, and then n=(x+1). Then the difference between the left & the right side is (x+n).

(n^2)-(x^2)=(x+n) there you go!
 
  • #5
Hi RobinSky! Welcome to PF! :smile:

I'm afraid epenguin already gave the answer, but apparently you haven't matched his formula with your own yet.

Cheers! :smile:
 
  • #6
Ahh yes, now I see it! Thanks anyway, fun though to see that I came up with this old formula in my own mind :biggrin:
 
  • #8
It is simply [itex](n+1)^2-n^2= n^2+ 2n+ 1- n^2= 2n+1[/itex]

Or, as epenguin did it, [itex](n+1)^2- n^2= (n+1+n)(n+1-n)= (2n+1)(1)= 2n+1[/itex].
 

1. What is the name of this formula?

The name of a formula is usually given based on its purpose or the variables involved in it. For example, the Pythagorean theorem is named after the Greek mathematician Pythagoras, who first discovered it.

2. How is this formula different from others?

A formula can be different from others in terms of the variables used, the conditions under which it is applicable, or the problem it solves. It is important to understand the specific characteristics and limitations of a formula to use it effectively.

3. Are there any alternative formulas for the same problem?

In many cases, there are multiple formulas that can be used to solve the same problem. It is important to compare and contrast different formulas to determine the most suitable one for a particular situation.

4. Can you explain the derivation of this formula?

The derivation of a formula involves understanding the logical steps and mathematical principles behind its development. It is important to understand the derivation of a formula to gain a deeper understanding of its applications and limitations.

5. How do I use this formula in real-life applications?

Many scientific formulas have practical applications in various fields such as engineering, physics, and chemistry. It is important to understand the real-life implications of a formula to effectively use it in problem-solving and decision-making.

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