Increase students' knowledge of patterns in Fibonacci numbers and practice pattern recognition.
Students will be able to continue patterns found, generalize patterns they see and find new patterns.
(These numbers get big, students may wish to use calculators.)
Add one more term to the sum of alternate terms of the Fibonacci numbers illustrated. That is, find the sum
a) Is this sum also a Fibonacci number?
b) Do the same thing for f(1)+f(3) +f(5) + f(7) +f(9) + f(11) +
f(13)+f(15).
c) Write down a rule that expresses the pattern illustrated.
d) Test your statement with 3 more examples.
Add one more term to the sum of alternate terms of the Fibonacci numbers illustrated. That is, find the sum
a) Are the sums related to Fibonacci numbers?
b) Write down a rule that expresses the pattern illustrated.
c) Test your statement with 3 more examples.
The result is 294 = f(2) + f(5) + f(10) + f(13).
a) Write 380 and 605 as sums of Fibonacci numbers.
Write the pattern found in these examples. Check with 2 more examples.
a) f(1)*f(3)-f(2)2 = ?
b) f(2)*f(4)-f(3)2 = ?
c) f(3)*f(5)-f(4)2 = ?
d) f(4)*f(6)-f(5)2 = ?
Show that the Lucas numbers l(5), l(6) and l(7) are each the sum of two Fibonacci numbers in the same way l(2), l(3) and l(4) were.
Another relationship: f(4)/f(2) = 3/1 = l(2); f(6)/f(3)= 8/2 = 4 = l(3)and
f(8)/f(4) = 21/3= 7= l(4). Write 3 more examples of this pattern.