Solve n+C_{2}=n(n+1)/2

$$n + C _ { 2 } = \frac { n ( n + 1 ) } { 2 }$$

Solve for C_2

$$C_{2}=-\frac{n\left(1-n\right)}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2n+2C_{2}=n\left(n+1\right)$$

Use the distributive property to multiply n by n+1.

$$2n+2C_{2}=n^{2}+n$$

Subtract 2n from both sides.

$$2C_{2}=n^{2}+n-2n$$

Combine n and -2n to get -n.

$$2C_{2}=n^{2}-n$$

Divide both sides by 2.

$$\frac{2C_{2}}{2}=\frac{n\left(n-1\right)}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$C_{2}=\frac{n\left(n-1\right)}{2}$$

Solve for n

$$n=\frac{-\sqrt{8C_{2}+1}+1}{2}$$

$$n=\frac{\sqrt{8C_{2}+1}+1}{2}\text{, }C_{2}\geq -\frac{1}{8}$$