Math 220.906
Writing Assignment #5
November 7, 2013
Due Thursday, November 14.
Problem C: We define the Fibonacci sequence {Fn }, for n ≥ 1 recursively as follows:
F1 = 1,
F2 = 1,
and for n ≥ 3,
Fn = Fn−1 + Fn−2 .
Thus the sequence starts out as
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . .
(a) Compute the sum
Pn
i=1
Fi for n = 1, 2, 3, 4, and 5. Use induction to show that
n
X
Fi = Fn+2 − 1.
i=1
(b) Compute the sum
n
X
Fi2 for n = 1, 2, 3, 4, and 5. Use induction to show that
i=1
n
X
Fi2 = Fn Fn+1 .
i=1
(c) For n ≥ 2, let
Tn = Fn2 − Fn+1 Fn−1 .
Calculate some values and conjecture a formula for Tn . Prove your formula by induc2
tion. (Hint: At some point it may be useful to use Fn+1
= Fn+1 (Fn + Fn−1 ).)