Question

The golden ratio rho can be defined by lim n → [infinity] bn = rho , where rho = 1 + 1/rho. Solve this equation for rho. (Round your answer to four decimal places.)

Answer 1

Answer:

$\\ \rho \approx 1.6180$ (rounding to four decimal places)

Step-by-step explanation:

As a clarification note: The Golden Ratio is usually represented by the Greek letter phi ($\phi$) and not by rho letter ($\rho$). To answer the question, we will use the rho letter.

To solve this equation:

$\\ \rho = 1 + \frac{1}{\rho}$

We need first to rearrange its terms and then apply the quadratic formula for it, since the result of such rearrangement is a quadratic equation.

Rearranging the formula

Then, the equation $\\ \rho = 1 + \frac{1}{\rho}$ can be multiplied by $\\ \rho$ to both of its sides. The equation remains the same in doing so.

$\\ \rho * (\rho = 1 + \frac{1}{\rho})$

$\\ \rho * \rho = \rho + \rho * \frac{1}{\rho}$

$\\ {\rho}^2 = \rho + \frac{\rho * 1}{\rho}$

$\\ {\rho}^2 = \rho + \frac{\rho}{\rho} * 1$

$\\ {\rho}^2 = \rho + 1 * 1$

$\\ {\rho}^2 = \rho + 1$

$\\ {\rho}^2 - \rho - 1 = 0$, which is a quadratic equation that can be solved using the well known quadratic formula aforementioned.

Solutions for the resulting equation

A quadratic equation is of the form:

$\\ a*x^2 + b*x + c = 0$

And the formula for solving it has two solutions:

$\\ x_{1} = \frac{-b + \sqrt{b^2 - 4*a*c}}{2*a}$

$\\ x_{2} = \frac{-b - \sqrt{b^2 - 4*a*c}}{2*a}$

Well, applying it for:

$\\ {\rho}^2 - \rho - 1 = 0$, we have $a = 1, b = -1, c = -1$

Then, the first solution is:

$\\ \rho_{1} = \frac{-(-1) + \sqrt{(-1)^2 - 4*1*(-1)}}{2*1}$

$\\ \rho_{1} = \frac{1 + \sqrt{(-1)(-1) - 4*1*(-1)}}{2*1}$

$\\ \rho_{1} = \frac{1 + \sqrt{1 + 4}}{2*1}$

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

The second solution is:

$\\ \rho_{2} = \frac{-(-1) - \sqrt{(-1)^2 - 4*1*(-1)}}{2*1}$

$\\ \rho_{2} = \frac{1 - \sqrt{1 + 4}}{2} = \frac{1 - \sqrt{5}}{2}$

But, the Golden Ratio is a positive number since it is a ratio between positive quantities, thus, the valid solution for the Golden Ratio is the first solution. The second solution is a negative number.

$\\ \rho_{1} = \frac{1 + \sqrt{5}}{2}$, which is the exact solution.

Since the question is asking to round this answer to four decimal places, then we have:

$\sqrt{5} = 2.2360679774997896964091...$

Therefore

$\\ \rho_{1} = \frac{1 + 2.2360679774997896964091...}{2}$

$\\ \rho_{1} \approx 1.61803398874989$ or

$\\ \rho_{1} \approx 1.6180$ (rounding to four decimal places).