Whakaoti i te left(5+5+left(n-1right)dright)n/2=390

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

Whakaoti mō d

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

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

Ngā Upane mō te Whakaoti Whārite Rārangi

Me whakarea ngā taha e rua ki te 2.

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

Tāpirihia te 5 ki te 5, ka 10.

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

Whakamahia te āhuatanga tohatoha hei whakarea te n-1 ki te d.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 10+nd-d ki te n.

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

Whakareatia te 390 ki te 2, ka 780.

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

Tangohia te 10n mai i ngā taha e rua.

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

Pahekotia ngā kīanga tau katoa e whai ana i te d.

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

Whakawehea ngā taha e rua ki te n^{2}-n.

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

Mā te whakawehe ki te n^{2}-n ka wetekia te whakareanga ki te n^{2}-n.

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

Whakawehe 780-10n ki te n^{2}-n.

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

Whakaoti mō n (complex solution)

$$\left\{\begin{matrix}n=\frac{\sqrt{d^{2}+3100d+100}+d-10}{2d}\text{; }n=\frac{-\sqrt{d^{2}+3100d+100}+d-10}{2d}\text{, }&d\neq 0\\n=78\text{, }&d=0\end{matrix}\right.$$

Whakaoti mō n

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