MathDB

5

Part of 2017 Irish Math Olympiad

Problems(2)

a_{n+2} = 2a_{n+1} + 41a_n

Source: Irish MO 2017 paper 1 problem 5

12/12/2022
The sequence a=(a0,a1,a2,...)a = (a_0, a_1,a_2,...) is defined by a0=0,a1=2a_0 = 0, a_1 =2 and an+2=2an+1+41ana_{n+2} = 2a_{n+1} + 41a_nProve that a2016a_{2016} is divisible by 2017.2017.
number theoryrecurrence relationalgebra
(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}

Source: Irish MO 2017 paper 2 problem 5

12/12/2022
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.
algebraSequenceSum