#### Activity Overview

The Golden Ratio and its association with the Fibonacci sequence is well known, but there is much more to explore. Variants of the Fibonacci sequence of the form t(n+2)=t(n)+a.t(n+1) also generate specific ratios. For a = 2 the ratio is referred to as the Silver Ratio, for a = 3 the bronze ratio, and in general, the metallic ratios. This activity explores these different ratios and compares the properties to the case where a = 1, the Golden Ratio.

#### Objectives

- Use recursion to generate a sequence
- Solve problems involving sequences
- Explore the most popular second order linear recursive relation

#### Vocabulary

- Recursion
- Fibonacci Sequence
- Golden Ratio
- Silver Ratio
- Bronze Ratio
- Metallic Ratios

#### About the Lesson

The Golden Ratio can be generated using consecutive terms of the classic form of the Fibonacci sequence: t(n+2)=t(n+1)+t(n). What if the Fibonacci ratio is altered slightly to: t(n+2)=a.t(n+1) + t(n) where 'a' is an integer? The ratio between consecutive terms also approaches limiting values for each 'a'. The classic Fibonacci sequence and Golden ratio align to a = 1. For the first variant a = 2 resulting in the Silver ratio and the second variant a = 3 results in the Bronze ratio and the subsequent metallic ratios. In this activity students generate each of these ratios, determine their exact values and check the corresponding properties against those of the Golden ratio.