When you add together three consecutive Fibonacci numbers the answer is double the largest number in the sequence e.g 1+1+2=4 and 34+55+89=178 this is because the two smallest numbers in the sequence add to make the biggest number in the sequence.
When you choose any four consecutive numbers, add together the first and last and divide by 2 the answer is always the second biggest in the sequence. E.g. 1,1,2,3 1+3 = 4 4 divided by 2 is 2. See attached photo.
When you add together any six consecutive Fibonnacci numbers and divide by 4 the answer is the number that is second biggest in the sequence e.g 1+1+2+3+5+8=20 and then 20 divided by 4 is 5. See attached photo. This is because the answer when you add is four times the second biggest number.