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−...

Welcome. So below you can see very beautiful painting of Friar Luca Pacioli. Luca Pacioli was a religious man but he was also a Mathematician . And because of his standing as mathematician he became the teacher and good friend of Leonardo Da Vinci. Who is the most famous of all the renaissance men . Luca Pacioli  Luca Pacioli is known as the Father of Accounting and Bookkeeping. But our interest in him is because he was the author of a book called the "De Divina Proportione"  or "On the Divine proportion". The Divine proportion is a number . It's a number that we call now the Golden Ratio. Luca Pacioli thought this number was divine, was godly number. We are going to see that this number has a very close relationship with Fibonacci Sequence. What Is GOLDEN RATIO ?? We can understand what the Golden Ratio is by starting with a line segment and dividing the line segment into two segments. Lets Say one of length x and one of length y . Assuming that length x is lar...

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...

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 ...

HEALING POWER OF NATURE N ature provides a bountiful (unbridge) pharmacopoeia through it's plant life. Plant were humankind's original medicines, and even with the birth of modern medicine plants remain an important source of medicinal help.  I n all traditional culture, certain members accrue ( emerge) specialized knowledge of medicinal plants and their applications. These are the healers, midwives, shamans, and other individual who dispence cures and remedies through their understanding of healing plants and their applications.  Women in their roles as mothers and grandmothers also command this knowledge, some times in the form of "Old wives"remedies that, more often than not, have a firm basis in herbal medicine.  T oday countless people still use medicinal plants, whether in traditional ways, in alternative and complementary medicine, or as building blocks for new research and innovative drugs. Pharmaceutical companies and ...

ELECTROCARDIOGRAM [ECG] Generally we all heared term "ECG" most of the time but most of you don't know what exact it is and why it is so important.  Let's know about it !  DISCOVERED by the Dutch Physiologist Einthoven willem( The father of ECG).  Electrocardiogram[ECG] is the graphical registration of electrical activities of the heart, which occure prior to the onset of mechanical activity of the heart.  The technique by which the electrical activities of the heart are studied called Electrocardiography and the instrument (machine) is called Electrocardiograph .  ECG is the summed electrical activity of all cardiac muscle fibers recorded from the body.  HOW IT IS RECORDED  The paper that is used for recording ECG is called ECG paper and Electrocardiographic grid refers to the markings on ECG paper.  ECG paper has horizontal and vertical lines at regular intervals of 1mm. •Time duration of ECG waves is plotted horizontally on X-axis...