Solve a_{n}=n^2+n+5/35n^2+400n-15

$$a _ { n } = \frac { n ^ { 2 } + n + 5 } { 35 n ^ { 2 } + 400 n - 15 }$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{\sqrt{162100a_{n}^{2}-160a_{n}-19}-400a_{n}+1}{2\left(35a_{n}-1\right)}\text{; }n=\frac{-\sqrt{162100a_{n}^{2}-160a_{n}-19}-400a_{n}+1}{2\left(35a_{n}-1\right)}\text{, }&a_{n}\neq \frac{1}{35}\\n=\frac{38}{73}\text{, }&a_{n}=\frac{1}{35}\end{matrix}\right.$$

Solve for a_n

$$a_{n}=\frac{n^{2}+n+5}{5\left(7n^{2}+80n-3\right)}$$

$$n\neq \frac{\sqrt{1621}-40}{7}\text{ and }n\neq \frac{-\sqrt{1621}-40}{7}$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\sqrt{162100a_{n}^{2}-160a_{n}-19}-400a_{n}+1}{2\left(35a_{n}-1\right)}\text{; }n=\frac{-\sqrt{162100a_{n}^{2}-160a_{n}-19}-400a_{n}+1}{2\left(35a_{n}-1\right)}\text{, }&a_{n}\leq -\frac{\sqrt{30863}}{16210}+\frac{4}{8105}\text{ or }\left(a_{n}\neq \frac{1}{35}\text{ and }a_{n}\geq \frac{\sqrt{30863}}{16210}+\frac{4}{8105}\right)\\n=\frac{38}{73}\text{, }&a_{n}=\frac{1}{35}\end{matrix}\right.$$