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science (Refraction of light)

Refraction of Light

Refraction of Light

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Fibonacci Sequence and Rabbit Problem

Hello friends hope you all are good , healthy and safe.  In this Blog we will learn about fibonacci Sequence , golden ratio,relation between them , rabbit problem and many more. Fibonacci  Greatest mathematician of the middle ages, Fibonacci. Fibonacci was born in pisa around 850 years ago. He was a famous Italian mathematician. In the year 1202 he finished his book "Liber Abeci". The book of calculations that brought Arabic numerals to Europe, the zero one,two that we use today. Fibonacci also presented a growing rabbit population problem. And after solving this problem he derived the famous sequence of numbers that is named after him, The Fibonacci Sequence . Suppose you put a male-female pair of newly born rabbits in a field . Rabbits take a month to mature before mating. After o ne month , females gives birth to one male-female pair and then mate again. No rabbits die. How many rabbit pairs are there after one year? To solve this, we construct table. At the start of each ...
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Fibonacci Sequence Problem set 1

 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           ...
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Fibonacci Sequence and Climbing Staircase problem

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...
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