SOLUTION: The arithmetic sequences 1, 5, 9, 13,... and 1, 8, 15, 22,... have infinitely many terms in common. Calculate the sum of the first three common terms.

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Question 1161201: The arithmetic sequences 1, 5, 9, 13,... and 1, 8, 15, 22,... have infinitely many terms in common. Calculate the sum of the first three common terms.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52817)   (Show Source): You can put this solution on YOUR website!
.

The common difference for the first AP is  4;  for the second AP  is  7.


The common terms form an AP with the first term 1 and the common difference equal to


    28 = 4*7 = Least Common Multiple of 4 and 7 = LCM(4,7).


Hence, the first three common terms are  1, 29, 57.


Their sum is  1 + 29 + 57 = 87.     ANSWER

Solved.


Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


The two arithmetic sequence have the term 1 in common.

The common differences in the two sequences are 4 and 7.

The sequence of terms that are common to both sequences will be an arithmetic sequence with a common difference of 28, which is the LCM of 4 and 7.

So the second term common to both sequences is 1+28 = 29.

The sum of the first three terms of the sequence of terms common to the two given sequences is 3 times the second term of the sequence.

ANSWER: 3*29 = 87
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