Finding the Fibonacci Numbers: A Similar Formula

In this series of posts about the Fibonacci sequence , a very famous sequence of numbers within mathematics, we have just concluded showing how you can take the recursive formula (which uses previous values of to compute the next values) and turn that formula into an exact formula that can skip right over the previousContinue reading “Finding the Fibonacci Numbers: A Similar Formula”

Finding the Fibonacci Numbers: The Formula

This is the third post in a series about an exact formula for the Fibonacci numbers, , which are defined by the initial values and the recurrence relation . We have made a lot of progress towards our goal. We discovered a connection between , the golden ratio , and the Fibonacci sequence by findingContinue reading “Finding the Fibonacci Numbers: The Formula”

The Quadratic Formula (Solution)

Having discussed the motivation of why something like a “quadratic formula” is a useful thing to discover and understand, I’d like to work through some of the ideas that might lead one to discover a quadratic formula. Reducing the Number of Unknowns Remember that the equation we care about is , with (so that thisContinue reading “The Quadratic Formula (Solution)”

Pythagorean Triples? (Solution #1, Part 2)

In Part 1, we have begun discussing primitive Pythagorean triples, and thought a little bit about them. Now, we want to try to characterize all primitive triples. That is now our goal. Limiting The Possibilities Suppose we are given a primitive triple (a,b,c). Recall that this means that the three positive whole numbers a, b,Continue reading “Pythagorean Triples? (Solution #1, Part 2)”

Pythagorean Triples? (Solution #1, Part 1)

(If you haven’t read the “Problem” post with the same title, go there first. This will make more sense if you do.) We want to find all the Pythagorean triples (a,b,c). The first thing a mathematician would probably do is to try some small examples, gather some information, and then look for patterns within thatContinue reading “Pythagorean Triples? (Solution #1, Part 1)”