Solve 64=n+n(n-1)/2d

$$64 = n + \frac { n ( n - 1 ) } { 2 } d$$

Solve for d

$$d=-\frac{2\left(n-64\right)}{n\left(n-1\right)}$$

$$n\neq 1\text{ and }n\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$128=2n+n\left(n-1\right)d$$

Use the distributive property to multiply n by n-1.

$$128=2n+\left(n^{2}-n\right)d$$

Use the distributive property to multiply n^{2}-n by d.

$$128=2n+n^{2}d-nd$$

Swap sides so that all variable terms are on the left hand side.

$$2n+n^{2}d-nd=128$$

Subtract 2n from both sides.

$$n^{2}d-nd=128-2n$$

Combine all terms containing d.

$$\left(n^{2}-n\right)d=128-2n$$

Divide both sides by n^{2}-n.

$$\frac{\left(n^{2}-n\right)d}{n^{2}-n}=\frac{128-2n}{n^{2}-n}$$

Dividing by n^{2}-n undoes the multiplication by n^{2}-n.

$$d=\frac{128-2n}{n^{2}-n}$$

Divide 128-2n by n^{2}-n.

$$d=\frac{2\left(64-n\right)}{n\left(n-1\right)}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{\sqrt{d^{2}+508d+4}-d+2}{2d}\text{; }n=-\frac{-\sqrt{d^{2}+508d+4}-d+2}{2d}\text{, }&d\neq 0\\n=64\text{, }&d=0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{\sqrt{d^{2}+508d+4}-d+2}{2d}\text{; }n=-\frac{-\sqrt{d^{2}+508d+4}-d+2}{2d}\text{, }&d\leq -96\sqrt{7}-254\text{ or }\left(d\neq 0\text{ and }d\geq 96\sqrt{7}-254\right)\\n=64\text{, }&d=0\end{matrix}\right.$$