# Monthly Archives: September 2007

## Explicit Fibonacci numbers

Don’t worry, this post isn’t going to be X-rated! By explicit I mean not recursive. Remember that the Fibonacci numbers are defined recursively, that is, each Fibonacci number is given in terms of previous ones: . Doesn’t it make you … Continue reading

Posted in famous numbers, fibonacci, golden ratio, proof | 4 Comments

## Golden powers

So, we know from a previous challenge that . That’s a pretty interesting property, which is shared only by its cousin, . I wonder whether other powers of have special properties too? Let’s see: Interesting! What about ? And ? … Continue reading

Posted in famous numbers, fibonacci, golden ratio, induction, proof | 6 Comments

## Challenge #10 Solution

Have you tried solving Challenge #10 yet? Go try it first if you haven’t. It’s not too hard, I promise!

Posted in famous numbers, golden ratio, proof, solutions | 3 Comments

## Golden ratio properties (Challenge #10)

Remember the golden ratio, (phi)? It’s the positive solution to the equation , which can be found using the quadratic formula: Closely related is its cousin, (phi-hat) . As we’ll see, these famous constants actually relate to Fibonacci numbers in … Continue reading