Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and...
Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and
fn+2=fn+1+fnfn+2=fn+1+fn for n∈N.
Prove that fn=[(1+√5)n−(1−√5)n]/2n√5.
1. Prove that an integer a is divisible by 5 if and only if a2
is...
1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...
Solution.The Fibonacci numbers are defined by the recurrence
relation is defined F1 = 1, F2 =...
Solution.The Fibonacci numbers are defined by the recurrence
relation is defined F1 = 1, F2 = 1 and for n > 1, Fn+1 = Fn +
Fn−1. So the first few Fibonacci Numbers are: 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, . . . There are numerous curious properties of
the Fibonacci Numbers Use the method of mathematical induction to
verify a: For all integers n > 1 and m > 0 Fn−1Fm + FnFm+1...