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Find all sequences that satisfy the following equation

Source: P6: BWM 2014

October 7, 2014

algebrabinomial theoremalgebra proposed

Problem Statement

For all positive integers

m

and

k

with

m\ge k

, define

a_{m,k}=\binom{m}{k-1}-3^{m-k}

. Determine all sequences of real numbers

\{x_1, x_2, x_3, \ldots\}

, such that each positive integer

n

satisfies the equation

a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0