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Limit of a sequence (nice, easy)

Source: IMC 2003 day 1 problem 1

October 14, 2005
limitIMCcollege contests

Problem Statement

(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
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