Question

how is the pascal triangle related to the fibonacci numbers with examples(100 words)​

Answer 1

Answer:

See the following diagram for what does this question mean.

Let Fibonacci number be Fn=Fn−1+Fn−2 , where F0=1,F1=1 .

Need to show ∑⌊n2⌋k=0(n−kn−2k)=Fn .

Use induction method.

When n=0 , ∑⌊n2⌋k=0(n−kn−2k)=(00)=F0 ; when n=1 , ∑⌊n2⌋k=0(n−kn−2k)=(11)=F1 .

When n is even,

Fn−2+Fn−1=∑n−22k=0(n−2−kn−2−2k)+∑n−22k=0(n−1−kn−1−2k)=

(n−1n−1)+∑n2−1k=1((n−1−kn−2k)+(n−1−kn−1−2k))+(n−220)=

(nn)+∑n2−1k=1(n−kn−2k)+(n20)= ∑n2k=0(n−kn−2k)=

∑⌊n2⌋k=0(n−kn−2k)=Fn

When n is odd,

Fn−2+Fn−1=∑n−12−1k=0(n−2−kn−2−2k)+∑n−12k=0(n−1−kn−1−2k)=

(n−1n−1)+∑n−12k=1((n−1−kn−2k)+(n−1−kn−1−2k))=

(nn)+∑n−12k=1(n−kn−2k)= ∑n−12k=0(n−kn−2k)=

∑⌊n2⌋k=0(n−kn−2k)=Fn