Question

Prove the following identities. (a) F1 +F3 +F5 +...+F2n−1 = F2n. (b) F0 −F1 +F2 −F3...

Prove the following identities. (a) F1 +F3 +F5 +...+F2n−1 = F2n. (b) F0 −F1 +F2 −F3 +...−F2n−1 +F2n = F2n−1 −1. (c) F02 +F12 +F2 +...+Fn2 = Fn ·Fn+1. (d) Fn−1Fn+1 − Fn2 = (−1)n.

Discrete math about Fibonacci numbers

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
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.
Recall that the Fibonacci numbers are defined by F0 = 0,F1 = 1 and Fn+2 =...
Recall that the Fibonacci numbers are defined by F0 = 0,F1 = 1 and Fn+2 = Fn+1 + Fn for all n ∈N∪{0}. (1) Make and prove an (if and only if) conjecture about which Fibonacci numbers are multiples of 3. (2) Make a conjecture about which Fibonacci numbers are multiples of 2020. (You do not need to prove your conjecture.) How many base cases would a proof by induction of your conjecture require?
The Fibonacci numbers are defined recursively as follows: f0 = 0, f1 = 1 and fn...
The Fibonacci numbers are defined recursively as follows: f0 = 0, f1 = 1 and fn = fn−1 + fn−2 for all n ≥ 2. Prove that for all non-negative integers n: fn*fn+2 = ((fn+1))^ 2 − (−1)^n
Let f1, f2, f3: [a,b] -->R be nonnegative concave functions such that f1(a) = f2(a) =...
Let f1, f2, f3: [a,b] -->R be nonnegative concave functions such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0. Suppose that max(f1) <= max(f2) <= max(f3). Prove that: max(f1) + max(f2) <= max(f1+f2+f3)
The Fibonacci series is given by; F0=0, F1=1,F2=1, F3=2,F4=3,…F(i)=F(i-1)+F(i-2) Given that r^2=r+1. Show that F(i) ≥...
The Fibonacci series is given by; F0=0, F1=1,F2=1, F3=2,F4=3,…F(i)=F(i-1)+F(i-2) Given that r^2=r+1. Show that F(i) ≥ r^{n-2}, where F(i) is the i th element in the Fibonacci sequence
The Fibonacci sequence is defined as follows F0 = 0 and F1 = 1 with Fn...
The Fibonacci sequence is defined as follows F0 = 0 and F1 = 1 with Fn = Fn−1 +Fn−2 for n > 1. Give the first five terms F0 − F4 of the sequence. Then show how to find Fn in constant space Θ(1) and O(n) time. Justify your claims
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...
The Fibonacci series can be defined recursively as: F1 = 0; F2 = 1; and Fn...
The Fibonacci series can be defined recursively as: F1 = 0; F2 = 1; and Fn = Fn-2 + Fn-1. Show inductively that: (F1)2 + (F2)2 + ... + (Fn)2 = (Fn)(Fn+1).
Three factories F1, F2 and F3 respectively produce 25%, 35% and 40% of the total number...
Three factories F1, F2 and F3 respectively produce 25%, 35% and 40% of the total number of electrical parts intended for the assembly of a machine. These factories respectively produce 1%, 2% and 3% of defective parts. We notice : The event A : "the part is produced by the F1 factory" The event B :  "the part is produced by the F2 factory" The event C : "the part is produced by the F3 factory" The event D : "the...
Determine if the set of functions is linearly independent: 1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x 2. f1(x)=e^...
Determine if the set of functions is linearly independent: 1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x 2. f1(x)=e^ x, f2(x)=e^-x, f3(x)=senhx
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT