# Category Archives: fibonacci

## A Fibonacci pattern

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences | Tagged , , | 4 Comments

## Math Teachers at Play #21

Math Teachers at Play #21 is up at Math Mama Writes…, and it includes this cute puzzle, which Sue apparently made up herself: The Numberland News runs personal ads. 21 was looking for a new friend and put an ad … Continue reading

## m-bracelets

It is easy to generalize number bracelets to moduli other than 10—at each step, add the two previous numbers and take the remainder of the result when divided by m. Here are some pretty pictures I made of the resulting … Continue reading

Posted in arithmetic, fibonacci, iteration, pattern, pictures, sequences | 6 Comments

## 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

## 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