Question

one of your curious students noted the following interesting relationship; when she added 2 consecutive odd integers, teh sum was divisible by 2. when she added 3 consecutive odd integers the sum was divisible by 3. she made the conjecture that the sum of n consecutive odd numbers is divisible by n. is she correct? how would you help her prove or disprove her conjecture

Answer 1

An even number is divisible by 2, so it can be represented by 2N, where N is an integer. If 1 is added to an even number, then it will become an odd number. Therefore, an odd number can be represented by 2N+ 1.

An odd number can be represented as 2N+1, then the succeeding odd number is 2N + 3 .So, the sum of two consecutive odd numbers can be found as

(2N+ 1) + (2N + 3) = 4N + 4.

This shows that the sum of two consecutive odd numbers is divisible by 2.

Similarly sum of three consecutive odd numbers,

(2N+1)+(2N+3)+(2N+5)=6N+9

And from above equation it is clear that sum of three consecutive odd numbers is divisible by 3.

Similarly the sum of n consecutive odd numbers can be given by,

(2N+1)+(2N+3)+(2N+5)+(2N+7)+(2N+9)+...........................(2N+2n-1)=n(2N+n)

which is divisible by n.