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Topic: Equal sums of terms in Arithmetic Progession
Replies: 1   Last Post: Dec 28, 2012 6:56 PM

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croomejohn@yahoo.co.uk

Posts: 1
Registered: 12/28/12
Equal sums of terms in Arithmetic Progession
Posted: Dec 28, 2012 5:18 PM
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1+2+...+14 = 15+16+...+20

1+4+7+10+13 = 16+19

1+6+11+...+251 = 256+261+...+356

1+7+13+...+55 = 61+67+73+79

So with d = 1,3,5 and 6 (and 10 and 16, I haven't shown these), there exists n and m such that

1 + (1+d) + ... + 1+ nd = 1+(n+1)d + ... + 1+ md

It is impossible with d = 2. Owing to 1+3+...+2n+1 = n^2 and 1+3+...2m+1 = m^2
so m^2 = 2n^2 leads to root(2) being rational m/n.




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