query: Prove $f_{1} + f_{3} + f_{5} +...+f_{2n-1} = f_{2n}$?
positive_passages.docid: 18461
positive_passages.text: Sum of odd Fibonacci Numbers
negative_passages.docid: 103706
negative_passages.text: If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
negative_passages.docid: 75448
negative_passages.text: Find all function satisfying $f(f(n))+f(n)=2n+3k$
negative_passages.docid: 112512
negative_passages.text: How to prove that there are finitely many $n$ such that $f(1+1/n) = \frac{1}{n+1}$?
negative_passages.docid: 91068
negative_passages.text: Proving that $f_2+f_4+\cdots+f_{2n}=f_{2n+1}-1$ for Fibonacci numbers by induction
negative_passages.docid: 31698
negative_passages.text: $f(m^2 + f(n)) = f(m)^2 + n$ for all natural numbers
negative_passages.docid: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
negative_passages.docid: 72557
negative_passages.text: Prove that $T(n) = T(n/3) + T2n/3) + 5n$ is $O(n log n)$
negative_passages.docid: 53024
negative_passages.text: $f(x)f(1/x)=f(x)+f(1/x)$
negative_passages.docid: 148330
negative_passages.text: Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 68528
negative_passages.text: $f(x)=x^n+5x^{n-1}+3$ can't be expressed as the product of two polynomials
negative_passages.docid: 14648
negative_passages.text: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer
negative_passages.docid: 102422
negative_passages.text: Irreducibly of $x^5-x-1$ over $\mathbb{F}_3$
negative_passages.docid: 155806
negative_passages.text: (Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$
negative_passages.docid: 73454
negative_passages.text: Prove that $f$ has finite number of roots
negative_passages.docid: 134885
negative_passages.text: proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $