Series of numbers and more (3 questions)

[amitr]

Hello. I'm trying something here...

I created the following formula for something I'm playing with:

x = ((a+9)*(a-2))/2 + b

There's a catch here, the following inequality has to be valid

a > b

Now I'm having some trouble here...

1) Is there some way to find a and b only having the value of x and knowing that inequality is always valid?

Let me elaborate more on this topic, I have a few more questions related to the same problem (even if it isn't obvious by just reading this topic, but it is all related):

This is probably basic math... so please bear with me.

I have the following sequence of numbers:

1 3 6 10 15...

It isn't an AP because the difference isn't constant.

It starts in 2, then 3, 4, 5 and so on. Changing 1 each time.

2) Is there a way to know if a number is in this sequence? For instance, is 55 in this sequence? (it is...)

3) What is the difference used to reach some number in the sequence? For instance, the number 55, what is the difference that was used to find it (it is 10, the previous number is 45).

 

[gunch]

The answer to the first is positive. Let x_{a,b} = \frac{(a+9)(a-2)}{2} + b, then,
x_{a+1,b} - x_{a,b} = a+4
Thus the interval [x_{a,0},x_{a,0}+a) consists of all elements you can get with a certain a, but any larger a will result in number greater than or equal to x_{a,0}+a+4 so by determing which x_{a,0} x lies above we can determine the value of a, and from that it's easy to get b. Of course there are also some invalid x that can never be obtained, namely those in an interval of the form [x_{a,0}+a,x_{a,0}+a+4).

As for question b, it's a famous sequence known as the triangular numbers. Every element is the sum of the first n natural numbers, so the elements are of the form \frac{n(n+1)}{2}. To determine whether a number x is in the sequence you can simply solve \frac{n(n+1)}{2} = x for n and see if you get a positive integral solution.
For future reference the following site is great for finding number sequences: http://www.research.att.com/~njas/sequences/index.html

You should be able to solve question 3 using the formula for triangular numbers.