Since f(1)=1, f(2)=1 then we have the Fibonacci numbers (but starting from n=1 not n=0): month. The number of young is the same as the top entry of the previous month which is also the total of the month-before-that, ie f(n-2). The number of mature pairs is the total of the previous month, ie f(n-1). If f(n) is the total number of pairs in month n, then it is the sum of the mature in month n plus the young in month n. the bottom entry is the TOP entry of the previous month TOTAL. The explicit formula to find the sum of the Fibonacci sequence of n terms is given by of the given generating function is the coefficient of i0n Fi Fn+2 - 1. the top entry is the TOTAL of the previous month young. These two processes are summarized in this diagram: The number of young in any month is just the number of pairs that were mature the previous month, so we just copy the top entry to the bottom of the next column. In month 4, all those that were "young" the previous month become "mature", together with those who were "mature" from the previous month, so the top entry is the sum of the two entries in the column of the previous month. Then Equation 1.6 gives a formula for the Fibonacci numbers in K. They give birth to a new pair in month 3. that and are distinct roots of the polynomial x2 x 1 in K. Explanation: The young pair are introduced into the field and are immature in month 1. Returning to Fibonacci's original rabbits problem, let's enumerate how many mature rabbit pairs there are each month and how many immature pairs (1 month old): month: 1 2 3 4 5 mature: 1 1 2 3. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham. The Fibonacci sequence is dened as follows: F0 0 F1 1 Fi Fi 1+Fi 2 i 2 (1) The goal is to show that Fn 1 p 5 pnqn (2) where p 1+ p 5 2 and q 1 p 5 2 : (3) Observe that substituting n 0, gives 0as per Denition 1 and 0as per Formula 2 likewise, substituting n 1, gives 1 from both and hence, the base cases hold. It is optional and only recommended for those who have used matrices before.Īll you need to know is what a matrix is and how they multiply. A proof using Matrices This proof uses matrices and gives a practical application of - or introduction to - eigenvalues and eigenlines.
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