Can you find a sequence of consecutive integers that add up to 1000?
This puzzle is based on the excellent book A First Step to Mathematical Olympiad Problems – which is full of problems that could be extended to become exploration ideas.
Step 1 – arithmetic formula
Our first step is to write out what we want:
a + (a+1) + (a+2) + … (a +n) = 1000
next we notice that the LHS is an arithmetic series with first term a, last term a+n and n+1 terms. Therefore we can use the sum of an arithmetic sequence formula:
Sn = 0.5n(u1 + un)
Sn = 0.5(n+1)(a + a+n) = 1000
Sn = (n+1)(2a+n) = 2000
Step 2 – logic
However, we currently have 2 unknowns, n and a, and only 1 equation – so we can’t solve this straight away. However we do know that both a and n are integers – and n can be taken as positive.
The next step is to see that one of the brackets (n+1)(2a+n) must be odd and the other even (if n is odd then 2a + n is odd. If n is even then n+1 is odd). Therefore we can look at the odd factors of 2000:
Step 3 – prime factorisation
Using prime factorisation: 2000 = 24 x 5³
Therefore any odd factors must solely come from the prime factor combinations of 5 – i.e 5, 25 and 125.
Step 4 – trial and error
So we now know that either (n+1) or (2a+n) must be 5, 25, 125. And therefore the other bracket must be 400, 80 or 16 (as 5 x 400 = 2000 etc). Next we can equate the (n+1) bracket to one of these 6 values, find the value of n and hence find a. For example:
If one bracket is 5 then the other bracket is 400.
So if (n+1) = 5 and (2a+n) = 400 then n = 4 and a = 198.
This means that the sequence: 198+199+200+201+202 = 1000.
If (n+1) = 400 and (2a+n) = 5 then n = 399 and a = -197.
This means the sequence: -197 + -196+ -195 … + 201 + 202 = 1000.
We follow this same method for brackets 25, 80 and 125,16. This gives the following other sequences:
28+29+30+…+51+52 = 1000
-54+-53+-52+…+69+70 = 1000
-27+-26+-25+…+51+52 = 1000
55+56+57+…+69+70 = 1000
So with a mixture of mathematical formulae, prime factorisation, logic and trial and error we have our solutions. A good example of how mathematics is often solved in reality!