I have three problems to answer dealing with Fibonacci numbers. I understand the first two, but would like for reassurrance that they are correct and lost on the last one and could use some help. Thank you.
1. Fibonacci numbers can not be used more than once. Find sums for numbers 75 and 40. 2+5+13+21+34=75
1+5+13+21=40
2. Writing the last equation in the Fibonacci
1=2-1
1+1=3-1
1+1+2=5-1
1+1+2+3=8-1
1+1+2+3+5=13-1
**1+1+2+3+5+8=21-1
*************************
1=1^2
1-1=-1^2
1-2+5=2^4
1-2+5-13=-3^2
1-2+5-13+34=5^2
**1-2+5-13+34-89=-8^2
This is the one I am completely lost on
3. Choose any term of the Lucas sequence other than the first and square it
1,3,4,7,11,18,29,47,76,123
Then multiply the terms immediately before and after the one you chose. Subtract the smaller result from the larger. Repeat this procedure two more times starting with different terms. Then make a conjecture about the result of the procedure.
I think I need to
3^2=9
9x1=3 9x4=36
36-3=33
*************************
4^2=16
16x3= 48 16x7=112
112-48=64
*************************
7^2=49
49x4=196 49x11=539
539-196=343
*************************
I do not know if this is correct or how to write a conjecture with the results. Appreciate all help
1,3,4,7,11,18,29,47,76,123
Pick number: 4, square it 16
then 3x7=21, 21-16=5
Pick number: 11, square it 121
then 7x18=126, 126-121= 5
pick number 47: square it 2209
then 29x76=2204, 2209-2204= 5
first two are correct.
1. Correct! The sums you found for numbers 75 and 40 using Fibonacci numbers are indeed correct. You properly added non-repeated Fibonacci numbers to get the desired sums.
2. It seems like you are attempting to write a pattern using the index of the Fibonacci numbers and their corresponding squares. However, your approach is incorrect, and the pattern you derived does not hold true for all Fibonacci numbers. Here's how you could correctly establish a pattern:
Write down a few Fibonacci numbers and their squares:
Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, ...
Squares: 1, 1, 4, 9, 25, 64, 169, 441, ...
From this, we can observe that the squares of Fibonacci numbers form a new sequence: 1, 1, 4, 9, 25, 64, 169, 441, ...
To generalize this pattern:
- The square of the n-th Fibonacci number is equal to the (n-1)-th Fibonacci number squared.
- In other words, (Fn)^2 = (Fn-1)^2
Now let's take a look at the conjecture you attempted for the Lucas sequence:
3^2 = 9
9 x 1 = 9
9 - 3 = 6 (not 33)
4^2 = 16
16 x 3 = 48
48 - 16 = 32 (not 64)
7^2 = 49
49 x 4 = 196
196 - 49 = 147 (not 343)
It seems like the approach you used does not yield consistent results, and your conjecture is not accurate based on the given calculations.
To make a conjecture about the result of the procedure for the Lucas sequence, you could continue testing more terms and observing the pattern in the differences between the products and the initial squared term. You can then propose a general rule or pattern.