(n(n+1)/2)=80200

Solution for (n(n+1)/2)=80200 equation:

(n(n+1)/2)=80200

We move all terms to the left:

(n(n+1)/2)-(80200)=0

We multiply all the terms by the denominator


(n(n+1)-80200*2)=0


We calculate terms in parentheses: +(n(n+1)-80200*2), so:

n(n+1)-80200*2

We add all the numbers together, and all the variables

n(n+1)-160400

We multiply parentheses

n^2+n-160400

Back to the equation:

+(n^2+n-160400)


We get rid of parentheses

n^2+n-160400=0

a = 1; b = 1; c = -160400;
Δ = b2-4ac
Δ = 12-4·1·(-160400)
Δ = 641601
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{641601}=801$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-801}{2*1}=\frac{-802}{2}
=-401
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+801}{2*1}=\frac{800}{2}
=400
$