Solve S_{n}=n/2(5n+9)

$$S _ { n } = \frac { n } { 2 } ( 5 n + 9 )$$

Solve for S_n

$$S_{n}=\frac{n\left(5n+9\right)}{2}$$

Solution Steps

Express \frac{n}{2}\left(5n+9\right) as a single fraction.

$$S_{n}=\frac{n\left(5n+9\right)}{2}$$

Use the distributive property to multiply n by 5n+9.

$$S_{n}=\frac{5n^{2}+9n}{2}$$

Divide each term of 5n^{2}+9n by 2 to get \frac{5}{2}n^{2}+\frac{9}{2}n.

$$S_{n}=\frac{5}{2}n^{2}+\frac{9}{2}n$$

Solve for n (complex solution)

$$n=\frac{\sqrt{40S_{n}+81}-9}{10}$$

$$n=\frac{-\sqrt{40S_{n}+81}-9}{10}$$

Solve for n

$$n=\frac{\sqrt{40S_{n}+81}-9}{10}$$

$$n=\frac{-\sqrt{40S_{n}+81}-9}{10}\text{, }S_{n}\geq -\frac{81}{40}$$