Solve N=3n(n+1)+1

$$N = 3 n ( n + 1 ) + 1$$

Solve for N

$$N=3n\left(n+1\right)+1$$

Solution Steps

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

$$N=3n^{2}+3n+1$$

Solve for n (complex solution)

$$n=\frac{\sqrt{12N-3}}{6}-\frac{1}{2}$$

$$n=-\frac{\sqrt{12N-3}}{6}-\frac{1}{2}$$

Solve for n

$$n=\frac{\sqrt{12N-3}}{6}-\frac{1}{2}$$

$$n=-\frac{\sqrt{12N-3}}{6}-\frac{1}{2}\text{, }N\geq \frac{1}{4}$$