Question

For the Golden Ratio ɸ, show that the following relationship holds: ɸ⁽ⁿ ⁺ ¹ ⁾= ɸⁿ + ɸ⁽ⁿ ⁺ ¹ ⁾ what is ɸ + ɸ^-1

Answer 1

The relationship ɸ⁽ⁿ ⁺ ¹ ⁾= ɸⁿ + ɸ⁽ⁿ ⁺ ¹ ⁾ holds for the Golden Ratio ɸ.

Step-by-step explanation:

To show that the relationship ɸ⁽ⁿ ⁺ ¹ ⁾= ɸⁿ + ɸ⁽ⁿ ⁺ ¹ ⁾ holds for the Golden Ratio ɸ, we can use the definition of the Golden Ratio itself. The Golden Ratio is the value that satisfies the equation ɸ = 1 + 1/ɸ. Let's substitute ɸ⁽ⁿ ⁺ ¹ ⁾ and ɸⁿ + ɸ⁽ⁿ ⁺ ¹ ⁾ into this equation and see if they are equal.

Starting with ɸ⁽ⁿ ⁺ ¹ ⁾:

ɸ⁽ⁿ ⁺ ¹ ⁾ = 1 + 1/ɸ⁽ⁿ ⁺ ¹ ⁾

We know that ɸ⁽ⁿ ⁺ ¹ ⁾ = ɸⁿ + ɸ⁽ⁿ⁺¹⁾, so we can substitute that into the equation:

ɸⁿ + ɸ⁽ⁿ ⁺ ¹ ⁾ = 1 + 1/(ɸⁿ + ɸ⁽ⁿ⁺¹⁾)

From here, we can simplify the equation algebraically and show that both sides of the equation are equal.