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

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

Solve for C_2

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

Solution Steps

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

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

Divide each term of n^{2}+n by 2 to get \frac{1}{2}n^{2}+\frac{1}{2}n.

$$C_{2}=\frac{1}{2}n^{2}+\frac{1}{2}n$$

Solve for n (complex solution)

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

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

Solve for n

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

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