Problem 54
Let \(\displaystyle u_n\) be the \(\displaystyle n\)-th Fibonacci number. Prove that \(\displaystyle \frac13\cdot1,6^n < u_n < 1,7^n. \) Difficulty: 3. |
Problem 55
Prove that any two consecutive Fibonacci-numbers are co-prime. Difficulty: 5. |
Problem 56
Prove that \(\displaystyle u_1^2 +\ldots +u_n^2 =u_n u_{n+1} .\) Difficulty: 5. |
Problem 52
Let \(\displaystyle u_n\) be the \(\displaystyle n\)-th Fibonacci number (\(\displaystyle u_0=0\), \(\displaystyle u_1=1\), \(\displaystyle u_2=1\), \(\displaystyle u_3=2\), \(\displaystyle u_4=3\), \(\displaystyle u_5=5\), \(\displaystyle u_6=8\), Difficulty: 6. |
Problem 53
Prove that \(\displaystyle u_n^2-u_{n-1}u_{n+1}=\pm1\). Difficulty: 6. |
Problem 57
Express the sums below in closed form!
Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |