n(n+1)/2=420

Solution for n(n+1)/2=420 equation:

n(n+1)/2=420

We move all terms to the left:

n(n+1)/2-(420)=0

We multiply all the terms by the denominator


n(n+1)-420*2=0

We add all the numbers together, and all the variables

n(n+1)-840=0

We multiply parentheses

n^2+n-840=0

a = 1; b = 1; c = -840;
Δ = b2-4ac
Δ = 12-4·1·(-840)
Δ = 3361
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{3361}}{2*1}=\frac{-1-\sqrt{3361}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{3361}}{2*1}=\frac{-1+\sqrt{3361}}{2}
$