Question

In: Advanced Math

For the Fibonacci sequence, prove the formula u2n+1 = un un+2 + (-1)n

For the Fibonacci sequence, prove the formula u²_n+1 = u_n u_n+2 + (-1)ⁿ

Solutions

Expert Solution

The Febonacci sequence is given by where .

We have to prove the formula

We shall prove this formula by using Strong Mathematical Induction on the set of natural numbers

for n = 1, the formula is

Also from the febonacci sequence we have

since

hence the formula is true for n = 1.

suppose the formula is true for n = 1,2,3,...,k

that is ........(2)

now take n = k+1,

from equation (1) we have

using equation (2)

using equation (1)

again using equation (1)

this is the formula for n = k+1

Hence the formula is true for n = k+1

Thus the formula is true for n = 1 and if we assume that it is true for n = 1,2,3,...,k, then it is true for n = k+1

therefore by using Mathematical induction we conclude that the formula is true for every .

Hence for the Febonacci sequence

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Fibonacci Sequence: F(0) = 1, F(1) = 2, F(n) = F(n − 1) + F(n −...

Fibonacci Sequence: F(0) = 1, F(1) = 2, F(n) = F(n − 1) + F(n − 2) for n ≥ 2 (a) Use strong induction to show that F(n) ≤ 2^n for all n ≥ 0. (b) The answer for (a) shows that F(n) is O(2^n). If we could also show that F(n) is Ω(2^n), that would mean that F(n) is Θ(2^n), and our order of growth would be F(n). This doesn’t turn out to be the case because F(n)...

(a) The Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence,...

(a) The Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and are characterised by the fact that every number after the first two is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 114, … etc. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. We define Fib(0)=0,...

Write the first four terms of the sequence defined by the recursive formula a1 = 2, an = an − 1 + n.

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove...

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove that a<=xn<=b for all n>=1 2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn - x_(n-1)| for n=2,3,..... 3. Prove {xn} is a cauchy sequence and hence is convergent Please show full working for 1,2 and 3.

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

The Fibonacci sequence is the series of numbers 0, 1, 1, 2, 3, 5, 8,.... Formally,...

The Fibonacci sequence is the series of numbers 0, 1, 1, 2, 3, 5, 8,.... Formally, it can be expressed as: fib0 = 0 fib1 = 1 fibn = fibn-1 + fibn-2 Write a multithreaded C++ program that generates the Fibonacci series using the pthread library. This program should work as follows: The user will enter on the command line the number of Fibonacci numbers that the program will generate. The program will then create a separate thread that will...

The Fibonacci sequence is the series of integers 0, 1, 1, 2, 3, 5, 8, 13,...

The Fibonacci sequence is the series of integers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 . . . See the pattern? Each element in the series is the sum of the preceding two elements. Here is a recursive formula for calculating the nth number of the sequence: Fib(N) = {N, if N = 0 or 1 Fib(N - 2) + Fib(N - 1), if N > 1 a) Write a recursive method fibonacci that returns...

Write a c++ program of the Fibonacci Sequence. Have the user enter a positive integer n...

Write a c++ program of the Fibonacci Sequence. Have the user enter a positive integer n and compute the nth Fibonacci number. The program should end when the user enters a number less than or equal to zero

how to find arithmetic sequence, geometric sequence, and fibonacci sequence on excel?

The Fibonacci Sequence is a series of integers. The first two numbers in the sequence are...

The Fibonacci Sequence is a series of integers. The first two numbers in the sequence are both 1; after that, each number is the sum of the preceding two numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... For example, 1+1=2, 1+2=3, 2+3=5, 3+5=8, etc. The nth Fibonacci number is the nth number in this sequence, so for example fibonacci(1)=1, fibonacci(2)=1, fibonacci(3)=2, fibonacci(4)=3, etc. Do not use zero-based counting; fibonacci(4)is 3, not 5. Your assignment...

Subjects