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...

  Mahesh Das lived in a village during the region of Emperor Akbar who lived in a Palace. One day, Emperor Akbar was riding in the forest admiring the beauty of nature and enjoying the wind. Suddenly, he realized that he lost his way back to the palace. He was feeling confused.  Mahesh Das was walking and saw Akbar. Akbar also saw him as well. Mahesh enquired and Akbar replied that he was lost. Mahesh consoled him. Akbar rode guided by Mahesh on foot. They went on until they saw the Palace far away. Akbar turned to Mahesh and gave him his ring. Akbar told him to come to his Palace, and whatever he wanted, he would reward him.  Mahesh agreed.  Several years later, when Mahesh became a man he remembered Akbar's promise of reward and went to the Palace. He walked towards the gate of the Palace and the gate guard saw him. When the guard questioned , Mahesh said he wanted to meet Emperor Akbar. The guard refused and tried to send him away. Mahesh insisted to stay and show...

  Right now in the year 2020, technology has advanced so much. We have smartphones, tablets and even lights that light up just by clapping your hands! Nowadays your smartphone can control your lights, TV. Even TVs have become super thin unlike the older bulky TVs. Technology has given us so many gadgets. Due to this our lives have become so easy with hardly any problems.  Replica of Thomas Edison's First Electric However hundreds of years ago life wasn't the same. Life was extremely difficult. Communication was a task. Letters took a long time to deliver and the reply took even longer. During day people worked but during the evening they were dependend on candle lamps. There were many problems. But one scientist’s inventions changed the world.  His name was Thomas Edison.   He was born in the USA and was known as America’s greatest inventor. Edison invented many things such as the gramophone, motion picture camera and his most important invention was the electri...

 A rich man had a son. The son again and again asked his father for a new car. However the father never agreed to buy him a car. After a few times, the son gave up and focussed on his studies. He then successfully graduated. His father called him to his room. The son wondered why he was summoned and walked to his room.  The father said,”Congratulations my son! You have graduated! I have a gift for you." The son was very excited thinking his father had finally bought him a car. The father gave the gift to the son. The son opened the box. Inside was a nice thick book with a leather cover. The son was disappointed because he imagined it to be a car.  The son got angry, threw the book away, packed his bag and left the house. The son worked hard and became successful and rich like his father. One day he wanted to meet his father and realised he must be old now.  He thought about forgetting what happened in the past and packed his bag to meet his father.  When he was ...

 One Day, a widow was walking on the road with her two children. The mother and children were begging for money and help.  On the way, they met a rich merchant and they begged him for food and money. The merchant said, “Send your children to me. I will teach them to turn wood into gold.”  The widow thought that the merchant was joking. Still she thought of sending her children with the merchant. The merchant taught the two children to make different types of furniture out of wood. After a few months the boys started making wooden furniture and earnt lot of money by selling it.  Within a few years they worked very hard and became very rich. The mother was shocked! She realised her sons became rich because the merchant had taught them to turn wood into gold.  Moral of the Story:  You need to remember two things to earn money. Hard work and Skill. With these two you will be able to earn a lot of money. These are very important and are the real gold. 

 A man was walking in his garden. While looking around he found a cocoon of a butterfly. The man found the cocoon very exciting and would watch it everyday. Then one day a small opening appeared on the cocoon and the butterfly was struggling to get out. The man became very excited and wanted to see the butterfly come out.  The butterfly was trying very hard and was almost goin to come out. The man was excited to see the butterfly, but the butterfly was stuck. The man saw that the butterfly was almost out of the cocoon but got stuck. The man kept wondering how he could help the butterfly as it was struggling a lot. He went inside his house and got a pair of scissors and snipped off the remaining part of the cocoon. The butterfly then emerged easily but its body was swollen and it has small shriveled wings. The butterfly was not able to fly and crawled on the ground.  The man failed to understand what happened. He did not hurt the butterfly. He merely helped by cutting the ...

We are responsible for martyred of soldiers of our country!!! I shame on myself and on all those people of India who are still buying Chinese products, after knowing  the ladhakh border situation, intrusion of Chinese soldiers,we don't have right to say ourselves a citizen of India.         Specially,I shame on all those celebrities who advertising Chinese  products in our country,whom which we trusted a most. The country of More than 1.33 billion population,We have lots of labour in our country,most of them  are illeterate , leaves there home town in search of work.Most of them don't know what to buy, they don't even know which products is Indian or Chinese . And  effect of buying such products of different companies specially Chinese. so whatever there favourite stars advertises they will buy regardless of which county or their effect. I am sorry for  these peoples because it's not their fault, fault is that we trusted on celebrity who is a...

Introduction       You have never seen dinosaur but imagine how they  looks like?  Just by reading name you got the  picture of it in your mind that's the power of imagination.  Therefore, Don't  just read imagine simultaneously.     Sometimes we comes to a turning point there Everything that happening its never happened  before....        Covid-19 a pendemic not declared as a pendemic until it's spread to whole world. Initially  corona virus widely spreading all over the world and it's reached our country too, so we came to the decision as lockdown for 21 days by the order of prime minister Narendra modi ji.    At that time our PM said the sentence he said: “If we are not able to adhere to this lockdown sincerely for 21 days, believe me, India will go back 21 years.”    Assume if we go back to 21 year what will be happen.....! Let's go to the flashback It was 1999,a fortunately recentl...