Solve left(5+5+left(n-1right)dright)n/2=390

$$\frac{ \left( 5+5+ \left( n-1 \right) d \right) n }{ 2 } =390$$

Solve for d

$$d=-\frac{10\left(n-78\right)}{n\left(n-1\right)}$$

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

Steps for Solving Linear Equation

Multiply both sides by 2.

$$\left(5+5+\left(n-1\right)d\right)n=390\times 2$$

Add 5 and 5 to get 10.

$$\left(10+\left(n-1\right)d\right)n=390\times 2$$

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

$$\left(10+nd-d\right)n=390\times 2$$

Use the distributive property to multiply 10+nd-d by n.

$$10n+dn^{2}-dn=390\times 2$$

Multiply 390 and 2 to get 780.

$$10n+dn^{2}-dn=780$$

Subtract 10n from both sides.

$$dn^{2}-dn=780-10n$$

Combine all terms containing d.

$$\left(n^{2}-n\right)d=780-10n$$

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

$$\frac{\left(n^{2}-n\right)d}{n^{2}-n}=\frac{780-10n}{n^{2}-n}$$

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

$$d=\frac{780-10n}{n^{2}-n}$$

Divide 780-10n by n^{2}-n.

$$d=\frac{10\left(78-n\right)}{n\left(n-1\right)}$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\sqrt{d^{2}+3100d+100}+d-10}{2d}\text{; }n=\frac{-\sqrt{d^{2}+3100d+100}+d-10}{2d}\text{, }&d\leq -20\sqrt{6006}-1550\text{ or }\left(d\neq 0\text{ and }d\geq 20\sqrt{6006}-1550\right)\\n=78\text{, }&d=0\end{matrix}\right.$$