How can I write a Fibonacci sequence using summation notation?

[dba]

Homework Statement

I have trouble with the summation notation.

\sum_{i=0}^{k}\binom{k}{i}f_{n+i}

How do I write this as a sequence based on the definition of Fibonacci sequence?

Homework Equations

Definition:
f(0)=0
f(1)=1
f(n)=f(n-1) + f(n-2) for n>=2

Example:
f(2) = f(1) + f(0) = 1+0 = 1
f(3) = f(2) + f(1) = 1+1 = 2
f(4) = f(3) + f(2) = 2+1 = 3
f(5) = f(4) + f(3) = 3+2 = 5
and so on

The Attempt at a Solution

I know how to write:

\sum_{i=1}^{n}(i) = 1+2+3+...+n

but I do not understand how to write the following Fibonacci sequence:

\sum_{i=0}^{k}\binom{k}{i}f_{n+i}

Can someone show me how to write this as an expanded version or give me an example how to do this?
Thank you.

 

[HallsofIvy]

\begin{pmatrix}n \\ i\end{pmatrix} is the "binomial coefficient"
\frac{n!}{i! (n-i)!}
\sum_{i=0}^k \begin{pmatrix}k\\ i \end{pmatrix}f_{n+i}= f_n+ kf_{n+1}+ (k(k-1)/2)f_{n+2}+ \cdot\cdot\cdot

So, for example, with k= 3
\sum_{i=0}^3\begin{pmatrix}3 \\ i\end{pmatrix}f_{n+i}= f_n+ 3f_{n+1}+ 3f_{n+2}+ f_{n+3}

 

[dba]

Thank you very much.
I understand this now