MathDB

1

Part of 1992 Hungary-Israel Binational

Problems(2)

a+1/a-2

Source: 3-rd Hungary-Israel Binational Mathematical Competition 1992

5/24/2007
Prove that if cc is a positive number distinct from 11 and nn a positive integer, then n2≤cn+c−n−2c+c−1−2.n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}.
inequalitiesinductionfunctioninequalities proposed
2^{j+1}|1+L_{2j} (Lucas sequence)

Source: 3-rd Hungary-Israel Binational Mathematical Competition 1992

5/24/2007
We examine the following two sequences: The Fibonacci sequence: F0=0,F1=1,Fn=Fn−1+Fn−2F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 } for n≥2n \geq 2; The Lucas sequence: L0=2,L1=1,Ln=Ln−1+Ln−2L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2} for n≥2n \geq 2. It is known that for all n≥0n \geq 0 Fn=αn−βn5,Ln=αn+βn,F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, where α=1+52,β=1−52\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}. These formulae can be used without proof. Prove that 1+L2j≡0(mod2j+1)1+L_{2^{j}}\equiv 0 \pmod{2^{j+1}} for j≥0j \geq 0.
modular arithmeticnumber theory proposednumber theory