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Part of 2012 Mediterranean Mathematics Olympiad

Problems(1)

Least alpha for which this sequence is positive

Source: Mediterranean MO 2012 Q1

For a real number

\alpha>0

, consider the infinite real sequence defined by

x_1=1

and \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. Determine the smallest

\alpha

for which all terms of this sequence are positive reals. (Proposed by Gerhard Woeginger, Austria)

algebra unsolvedalgebra