Testing for Triangular NumbersDate: 06/09/99 at 23:24:19 From: agnes Subject: How to know if 12456, 1225, 13456 are triangular or sq. no. Doctor, please help me with this; I am really stuck. How do I figure out whether 12456, 1225, and 13456 are triangualar or square numbers? I know that the formula of a triangualar number is n(n+1)/2, but I don't know how it works. Agnes Date: 06/10/99 at 16:46:31 From: Doctor Terrel Subject: Re: How to know if 12456, 1225, 13456 are triangular or sq. no. Dear Agnes, In order to know if a number is triangular or not, it's helpful to know the formula for triangular numbers: n(n+1) T = -------- 2 Given a number to test, I'd do it this way: For example, 66. 66 = n(n+1)/2 132 = n(n+1) Now I reason that n and (n+1) are numbers relatively close together. So if I found the square root of 132, it should be close to the smaller "n" . Sure enough, sqrt(132) = 11.xxxxx. So if I now use 11 in the formula for "n" I have 11(11 + 1)/2 = 11(12)/2 = 11(6) = 66. And it works! If another number, like 100, was tested in this way, things don't work out. 100 = n(n+1)/2 200 = n(n+1) sqrt(200) = 14.xxxx But 14(15)/2 = 105, which does not equal 100. etc. Okay? Testing for squares should be easy on a calculator, if the number isn't too large of course. But do remember one simple fact: if the number ends with a 2, 3, 7, or 8, it CANNOT be square, no matter what. Do you see why? - Doctor Terrel, The Math Forum http://mathforum.org/dr.math/ Date: 11/04/2001 at 19:46:41 From: Paul Schuler Subject: Triangular numbers Is there a formula to calculate the following: A person enters a number. You need to find out if the number entered is the triangular value of another integer. For example, if a person enters the number 15, the formula would return the value 5. I know how to calculate the triangular value n(n+1)/2; I just can't see how to reverse this. Thank you. Date: 11/05/2001 at 05:08:06 From: Doctor Floor Subject: Re: Triangular numbers Hi, Paul, Thanks for writing. Dr. Terrel's answer does not fully satisfy me. Suppose we have a triangular number: n(n+1) T = ------ 2 then this is equivalent to 8T = 4n^2 + 4n and 8T + 1 = 4n^2 + 4n + 1 = (2n + 1)^2 So a number T is triangular if and only if 8T + 1 is an odd perfect square. If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ Date: 07/21/2001 at 20:09:19 From: Matt Subject: Testing for triangular numbers Please tell me the formula to test for triangular numbers - e.g., Is there a formula to test if X is a triangular number? Thanks, Matt Date: 07/23/2001 at 17:20:45 From: Doctor Pete Subject: Re: Testing for triangular numbers Hi, A triangular number has the form n(n+1)/2, for some integer n > 0. So, to see if a number x is triangular, let x = n(n+1)/2. Solving for x, we obtain n^2 + n - 2x = 0 or n = (Sqrt[1+8x] - 1)/2. Note n must be an integer greater than 0. Therefore, 1+8x must be an odd perfect square, and as you can see, whenever it is, we get a value of n that "works." So the test to see if x is triangular is to calculate 1+8x and see if it is odd, and a perfect square. - Doctor Pete, The Math Forum http://mathforum.com/dr.math/ |
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