Innovious 360

 Problems  1. The Fibonacci numbers can be extended to zero and negative indices using the relation Fn = Fn+2 − Fn+1 . Determine F0 and find a general formula for F−n in terms of Fn. Prove your result using mathematical induction. Solution: F0 = F2 − F1 = 0, F−1 = F1 − F0 = 1, F−2 = F0 − F−1 = −1, F−3 = F−1 − F−2 = 2, F−4 = F−2 − F−3 = −3, F−5 = F−3 − F−4 = 5, F−6 = F−4 − F−5 = −8. The correct relation appears to be F−n = (−1)^(n+1)×Fn                .............(1) We now prove equation (1) by mathematical induction. Base case: Our calculation above already shows that equation (1) is true for n = 1 and n = 2, that is, F−1 = F1 and F−2 = −F2. Induction step: Let us  assume that (1) is true for positive integers n = k − 1 and n = k. Then we have F−(k+1) = F−(k−1) − F(−k)     ..(from definition)                = (−1)^k×Fk−1 − (−1)^(k+1)×Fk           ...

Welcome. In this blog I will tell you relation between the Fibonacci Sequences and the Golden Ratio.Before that below  you can see the statue of Fibonacci, made in 1863 by Giovanni Paganucci, a sculpture in florence, but kept in a ancient cemetery in pisa where the Fibonacci was born. Its is interesting that the likeness of Fibonacci in this statue and his iconic portrait probably looks nothing like Fibonacci, since no true drawings of him exist from 850 years ago . But nevertheless Italy still honors him with this sculpture.  Statue Of Fibonacci Let's Return to Fibonacci Numbers and Fibonacci recursion relation. Then we will show you how they are related to Golden Ratio. So let's do some mathematics. Any way what's the recursion relation do you remember?? The n+1 Fibonacci number is equal to the sum of preceding two that is nth Fibonacci number Plus n-1 Fibonacci number right .                         Fn+1 = Fn + Fn−...

Hello Friends, So here is our next blog on Fibonacci  Sequence. In this blog I will introduce another problem whose solution is the Fibonacci Number. This problem is known as Climbing Staircase problem. The Question is How many ways one can climb staircase with n steps, taking one or two steps at a time ?? Questions Of Climbing Staircase  Eg. Suppose we have 3 steps to climb. So we have a Staircase, we climb it by taking one step , one step , one step or two step , one step or one step, two step. So If we have n-steps in Staircase , then how many different ways can we climb the Staircase? So to answer this question we could make a table, considering small numbers of steps i.e n = 1,2,3,4,5 And we can list the number of ways to climb the Staircase. So Observe the table first column is number of stairs or steps. Second column is the list of ways one can climb by taking one or two steps at a time. Third column a n   the total number of ways to climb Staircase. I hope y...