Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A is a set of all natural numbers less than $500$ which can be written as sum of two or more consecutive natural numbers.How to find the maximum possible cardinality of A?

share|improve this question

2 Answers 2

The only numbers that can't be written as a sum of two or more consecutive natural numbers are the powers of 2. So there is no "maximum possible" cardinality of $A$. Just take out the powers of 2 that are less than 500, and the numbers 0 and 1.

Proof: If $n=k+(k+1)+...+k+(l-1)=lk+\frac{1}{2}(l-1)l$ then $2n=2lk+l(l-1)=l(2k+l-1)$. If one of these factors is even, the other one is odd. So it all comes down to which numbers can be factored as a product of an even number and and odd number, and this leads to my claim.

share|improve this answer

Any odd number is of the form $2k+1=k+(k+1)$ so can be written as a sum of two consecutive numbers. How many of them are less than $500$? Any number that is a multiple of $3$ is of the form $3k=(k-1)+k+(k+1)$ How many of these are there? You can continue like this-sums of four successive numbers will be related to multiples of $4$ (how?) and so on. Then you have to worry about double counting. For example $15=7+8=4+5+6=1+2+3+4+5$

Alternately, you can just make a list of all the ones you can express with two numbers, all you can express with three, and so on, then throw out the duplicates.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.