Total induction out of number

Question

If you have a variable x which is the outcome of a total induction of a real number, what is the formula to find out which induction it was.

For example:

x = 6: 3

x = 10: 4

x = 15: 5

x = 21: 6

Answer 1

The terminology in the question is unfamiliar to me, but I think what it's asking is: suppose $x=1+\cdots+n$; then what is $n$ in terms of $x$?

Well, there's a nice simple formula: $x=\frac{n(n+1)}2$. Rearrange this a little to put it in the usual form for a quadratic equation in $n$: $n^2+n-2x=0$. And apply the usual quadratic formula: $n=\frac{\sqrt{8x+1}-1}2$. (The quadratic equation has two solutions but you evidently want the positive one.)

For instance, if $x=15$ this becomes $n=\frac{\sqrt{121}-1}2=\frac{11-1}2=5$.