Solve n(n+1)/2=2012

$$\frac { n ( n + 1 ) } { 2 } = 2012$$

Solve for n

$$n = \frac{\sqrt{16097} - 1}{2} \approx 62.936976599$$

$$n=\frac{-\sqrt{16097}-1}{2}\approx -63.936976599$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$n\left(n+1\right)=2012\times 2$$

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

$$n^{2}+n=2012\times 2$$

Multiply 2012 and 2 to get 4024.

$$n^{2}+n=4024$$

Subtract 4024 from both sides.

$$n^{2}+n-4024=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -4024 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-4024\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-4024\right)}}{2}$$

Multiply -4 times -4024.

$$n=\frac{-1±\sqrt{1+16096}}{2}$$

Add 1 to 16096.

$$n=\frac{-1±\sqrt{16097}}{2}$$

Now solve the equation n=\frac{-1±\sqrt{16097}}{2} when ± is plus. Add -1 to \sqrt{16097}.

$$n=\frac{\sqrt{16097}-1}{2}$$

Now solve the equation n=\frac{-1±\sqrt{16097}}{2} when ± is minus. Subtract \sqrt{16097} from -1.

$$n=\frac{-\sqrt{16097}-1}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{16097}-1}{2} n=\frac{-\sqrt{16097}-1}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$n\left(n+1\right)=2012\times 2$$

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

$$n^{2}+n=2012\times 2$$

Multiply 2012 and 2 to get 4024.

$$n^{2}+n=4024$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=4024+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=4024+\frac{1}{4}$$

Add 4024 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{16097}{4}$$

Factor n^{2}+n+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{16097}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{16097}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{\sqrt{16097}}{2} n+\frac{1}{2}=-\frac{\sqrt{16097}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=\frac{\sqrt{16097}-1}{2} n=\frac{-\sqrt{16097}-1}{2}$$