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First Term Finding (Posted on 2016-05-04) Difficulty: 3 of 5
Consider two arithmetic sequences having the common first term as A, where A is an integer. All the terms of the two sequences are integers.

Determine the minimum absolute value of A such that:

The 20th term of the first sequence equals the 16th term of the second sequence, and:

The sum of the first 20 terms of the first sequence equals the sum of the first 16 terms of the second sequence.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 2
Let X and Y be the difference between terms in the first and second sequences, respectively.  Since A and all terms of both sequences are integers, X and Y are integers as well. 

From the first statement, we know that (A + 19X) = (A + 15Y), so X = 15Y/19.

Ignoring the trivial solution of A = X = Y = 0, possible integer pairs (X,Y) are then (15,19), (30,38), etc.

From the second statement we know that 10(2A + 19X) = 8(2A + 15Y).  Substituting for X and simplifying gives A = (-15/2)Y, thus Y must be even (since A is an integer).  

So the minimal (absolute value) solution is when X = 30 and Y = 38, which gives A = -285.  

  Posted by tomarken on 2016-05-04 12:46:46
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