Question

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
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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
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4^2=16
16x3= 48 16x7=112
112-48=64
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7^2=49
49x4=196 49x11=539
539-196=343
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I do not know if this is correct or how to write a conjecture with the results. Appreciate all help

Answer 1

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.