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(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}

Source: Irish MO 2017 paper 2 problem 5

December 12, 2022
algebraSequenceSum

Problem Statement

Given a positive integer mm, a sequence of real numbers a=(a1,a2,a3,...)a= (a_1,a_2,a_3,...) is called mm-powerful if it satisfies (∑k=1nak)m=∑k=1nakm(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}for all positive integers nn. (a) Show that a sequence is 3030-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is 20172017-powerful.
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