100n(n+1)/2=10000

Solution for 100n(n+1)/2=10000 equation:

100n(n+1)/2=10000

We move all terms to the left:

100n(n+1)/2-(10000)=0

We multiply all the terms by the denominator


100n(n+1)-10000*2=0

We add all the numbers together, and all the variables

100n(n+1)-20000=0

We multiply parentheses

100n^2+100n-20000=0

a = 100; b = 100; c = -20000;
Δ = b2-4ac
Δ = 1002-4·100·(-20000)
Δ = 8010000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{8010000}=\sqrt{90000*89}=\sqrt{90000}*\sqrt{89}=300\sqrt{89}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(100)-300\sqrt{89}}{2*100}=\frac{-100-300\sqrt{89}}{200}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(100)+300\sqrt{89}}{2*100}=\frac{-100+300\sqrt{89}}{200}
$