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

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

Solve for a_n

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

Solution Steps

Use the distributive property to multiply n by 2n+1.

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

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

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

Solve for n

$$n=\frac{\sqrt{16a_{n}+1}-1}{4}$$

$$n=\frac{-\sqrt{16a_{n}+1}-1}{4}\text{, }a_{n}\geq -\frac{1}{16}$$