The number 1000 can be written in several ways as a sum of one or more consecutive positive integers, for instance, 1000=1000 (one summand) or 1000=198+199+200+201+202 (five summands). Find the largest possible number of summands in a representation of 1000 as a sum of consecutive positive integers.

Answer 1

If there are k summands, starting at n, we want n as small as possible.

n+(n+1)+...+(n+k) = 1000
kn + k(k-1)/2 = 1000
k^2 + (2n-1)k - 2000 = 0
so
4n^2-4n+8001 is a perfect square

If n=28, 4n^2+4n+8001=105^2 and k=25

check:
summing the arithmetic sequence starting at 28 for 25 terms,

S25 = 25/2 (28*2+24) = 25/2*80 = 1000

So, it looks like 25 is the maximum number of summands.