Suppose I start with a number 52, and then add to it 3... and then to that, 3+3... and to that, 3+3+3, etc... such as:
52, 55, 61, 70, 82, etc.
Now with that going on indefinitely, would I be able to determine whether 3,132 fell on that curve somewhere without using iteration? If so, what formula would I use?
It would have to take into account the starting point (52), the curve (3) [note: this number would always repeat itself in this situation, ie. 2=2,2+2,2+2+2,2+2+2+2, 3=3,3+3,3+3+3,3+3+3+3, etc.], and the number we want to check (3,132).
Hopefully I've explained this properly. Thanks!
You're trying to define a sequence $a(n)$ for $n=0,1,2,3,...$. This sequence follows the following recurrence : $$ a(n) = a(n-1) + n\cdot d $$ where $d$ is the "step".
Giving a "starting point" $s$ like $52$ you get the general formula : $$ a(n) = s+ \frac{1}{2} d \cdot n (n+1)\ \ \ \ \ \text{for } n = 0,1,2,3,... $$ So with your first example you put $s = 52,\ d = 3$ and get : $a(n) = 52 + \frac{3}{2} n(n+1)$
Then you just need to solve $a(n) = \text{your target number}$. In your example : $$ 52 + \frac{3}{2} n(n+1) = 3132 $$ which yields a non-integer solution for $n$ so $3132$ will never be generated by your sequence.