Calvin Lin inspired me to this problem.In his problem Arithmetic Progressions.I tried to generalize the problem taking it as 'How many arithmetic progressions of length are there, such that the terms are integers from to ?'
I found that when common difference is ,total A.Ps will be
When common difference is ,total A.Ps will be
The common difference will be continued till
{ where is greatest integer contained in ]
Let .
So I found that the total number of arithmetic progressions will be
For example
So the generalization formula will be
Please my friends if you find this useful Re-share and like.If you find any fault please tell me.
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There is a very simple way to obtain the total number of AP's of a fixed length t.
Hint: What can you say about at−a1?
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Thank you for you hint but sir can you please define at and a1 here?
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The first and last term of the arithmetic progression of length t.