A maths problem from 1988
Can someone help me find a solution?
A sequence a(i) with all a(i) in {1,2,...,n-1} is such that
a(1) + ... + a(n) < 2n (and obviously the sum is also at least n)
prove that there is a subsequence with sum n for all natural numbers n.
It has been troubling me for a while now and I started doubting whether or not I have the resources to solve it since the syllabus might have been different in 1988.
A sequence a(i) with all a(i) in {1,2,...,n-1} is such that
a(1) + ... + a(n) < 2n (and obviously the sum is also at least n)
prove that there is a subsequence with sum n for all natural numbers n.
It has been troubling me for a while now and I started doubting whether or not I have the resources to solve it since the syllabus might have been different in 1988.
Original post by Evariste18
Can someone help me find a solution?
A sequence a(i) with all a(i) in {1,2,...,n-1} is such that
a(1) + ... + a(n) < 2n (and obviously the sum is also at least n)
prove that there is a subsequence with sum n for all natural numbers n.
It has been troubling me for a while now and I started doubting whether or not I have the resources to solve it since the syllabus might have been different in 1988.
A sequence a(i) with all a(i) in {1,2,...,n-1} is such that
a(1) + ... + a(n) < 2n (and obviously the sum is also at least n)
prove that there is a subsequence with sum n for all natural numbers n.
It has been troubling me for a while now and I started doubting whether or not I have the resources to solve it since the syllabus might have been different in 1988.
Induction looks like the way to go here. Con you solve the problem when n=1? Can you do it when n=2?
Never mind, I solved it using the stonger statement that for each n there is a subsequence with sum k for all values of k in {1,...,n} instead of just n and then proceding by induction. The problem is just difficult to solve when one does not assume the stronger statement.
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