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.
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.
To find the largest possible number of summands in a representation of 1000 as a sum of consecutive positive integers, we need to think about the factors of 1000.
The prime factorization of 1000 is 2^3 * 5^3.
To find the largest possible number of summands, we want to divide 1000 into as many equal parts as possible.
Since the prime factors of 1000 are 2 and 5, we need to determine how many powers of 2 and 5 we can include.
For the power of 2:
2^0 = 1 (1 summand)
2^1 = 2 (2 summands)
2^2 = 4 (4 summands)
2^3 = 8 (8 summands)
Similarly, for the power of 5:
5^0 = 1 (1 summand)
5^1 = 5 (5 summands)
5^2 = 25 (25 summands)
To determine the largest possible number of summands for 1000, we should consider the product of the largest powers of 2 and 5 that we can include.
The largest power of 2 is 2^3 = 8.
The largest power of 5 is 5^2 = 25.
So, the largest possible number of summands for 1000 is (8 * 25) = 200.
Therefore, we can represent 1000 as a sum of 200 consecutive positive integers.