Solve 1/1+sqrt{5/2}+1

$$\frac{ 1 }{ \frac{ 1+ \sqrt{ 5 } }{ 2 } } +1$$

Evaluate

$$\frac{\sqrt{5}+1}{2}\approx 1.618033989$$

Solution Steps

Divide 1 by \frac{1+\sqrt{5}}{2} by multiplying 1 by the reciprocal of \frac{1+\sqrt{5}}{2}.

$$\frac{2}{1+\sqrt{5}}+1$$

Rationalize the denominator of \frac{2}{1+\sqrt{5}} by multiplying numerator and denominator by 1-\sqrt{5}.

$$\frac{2\left(1-\sqrt{5}\right)}{\left(1+\sqrt{5}\right)\left(1-\sqrt{5}\right)}+1$$

Consider \left(1+\sqrt{5}\right)\left(1-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(1-\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}+1$$

Square 1. Square \sqrt{5}.

$$\frac{2\left(1-\sqrt{5}\right)}{1-5}+1$$

Subtract 5 from 1 to get -4.

$$\frac{2\left(1-\sqrt{5}\right)}{-4}+1$$

Divide 2\left(1-\sqrt{5}\right) by -4 to get -\frac{1}{2}\left(1-\sqrt{5}\right).

$$-\frac{1}{2}\left(1-\sqrt{5}\right)+1$$

Use the distributive property to multiply -\frac{1}{2} by 1-\sqrt{5}.

$$-\frac{1}{2}-\frac{1}{2}\left(-1\right)\sqrt{5}+1$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$-\frac{1}{2}+\frac{1}{2}\sqrt{5}+1$$

Convert 1 to fraction \frac{2}{2}.

$$-\frac{1}{2}+\frac{1}{2}\sqrt{5}+\frac{2}{2}$$

Since -\frac{1}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

$$\frac{-1+2}{2}+\frac{1}{2}\sqrt{5}$$

Add -1 and 2 to get 1.

$$\frac{1}{2}+\frac{1}{2}\sqrt{5}$$