Solve S_{n}=3n(n+1)/2

$$S _ { n } = \frac { 3 n ( n + 1 ) } { 2 }$$

Solve for S_n

$$S_{n}=\frac{3n\left(n+1\right)}{2}$$

Solution Steps

Use the distributive property to multiply 3n by n+1.

$$S_{n}=\frac{3n^{2}+3n}{2}$$

Divide each term of 3n^{2}+3n by 2 to get \frac{3}{2}n^{2}+\frac{3}{2}n.

$$S_{n}=\frac{3}{2}n^{2}+\frac{3}{2}n$$

Solve for n (complex solution)

$$n=\frac{\sqrt{24S_{n}+9}}{6}-\frac{1}{2}$$

$$n=-\frac{\sqrt{24S_{n}+9}}{6}-\frac{1}{2}$$

Solve for n

$$n=\frac{\sqrt{24S_{n}+9}}{6}-\frac{1}{2}$$

$$n=-\frac{\sqrt{24S_{n}+9}}{6}-\frac{1}{2}\text{, }S_{n}\geq -\frac{3}{8}$$