MathDB

1

Part of 2003 IMC

Problems(2)

Limit of a sequence (nice, easy)

Source: IMC 2003 day 1 problem 1

10/14/2005
(a) Let a1,a2,...a_1,a_2,... be a sequenceof reals with a1=1a_1=1 and an+1>32ana_{n+1}>\frac32 a_n for all nn. Prove that lim⁡n→∞an(32)n−1\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}} exists. (finite or infinite) (b) Prove that for all α>1\alpha>1 there is a sequence a1,a2,...a_1,a_2,... with the same properties such that lim⁡n→∞an(32)n−1=α\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha
limitIMCcollege contests
Commuting

Source: IMC 2003 day 2 problem 1

10/14/2005
Let A,B∈Rn×nA,B \in \mathbb{R}^{n\times n} such that AB+B+A=0AB+B+A=0. Prove that AB=BAAB=BA.
linear algebramatrixlinear algebra unsolved