Proof involving fibonacci number: $ \binom{n}{1}F_1+\binom{n}{2}F_2+\binom{n}{3}F_3+\cdots+\binom{n}{n-1}F_{n-1}+F_n=F_{2n}, $
133150
Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$
73191
Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$
14784
Inductive proof of the closed formula for the Fibonacci sequence
49066
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
74887
Combinatorial proof of the identity ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+...+{n-1\choose k}+{n\choose k}$
4855
How to show that this binomial sum satisfies the Fibonacci relation?
131400
Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
27217
For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$
139426
Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction
59612
Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
42105
Fibonacci proof by induction
156566
Combinatorial proof of a binomial identity: $\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$
59638
How to prove this result on binomial coefficients?
106502
Prove by induction that for the Fibonacci numbers $F(n)$ with $n \ge 6$, $F(n) \ge 2^{n/2}$
145524
Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
124917
Fibonacci numbers and proof by induction