(n(n+1)/2)=2190

Solution for (n(n+1)/2)=2190 equation:

(n(n+1)/2)=2190

We move all terms to the left:

(n(n+1)/2)-(2190)=0

We multiply all the terms by the denominator


(n(n+1)-2190*2)=0


We calculate terms in parentheses: +(n(n+1)-2190*2), so:

n(n+1)-2190*2

We add all the numbers together, and all the variables

n(n+1)-4380

We multiply parentheses

n^2+n-4380

Back to the equation:

+(n^2+n-4380)


We get rid of parentheses

n^2+n-4380=0

a = 1; b = 1; c = -4380;
Δ = b2-4ac
Δ = 12-4·1·(-4380)
Δ = 17521
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{17521}}{2*1}=\frac{-1-\sqrt{17521}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{17521}}{2*1}=\frac{-1+\sqrt{17521}}{2}
$