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Question 1150161: The sum of 18 consecutive odd integers is a perfect fifth power of n. If x is the smallest possible first number of the series, then what is the product of nx?
Answer by greenestamps(13200)   (Show Source):
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Any 18 consecutive odd integers form an arithmetic sequence.
The sum of an arithmetic sequence is the number of terms, multiplied by the average of the terms.
Let the average of the 18 terms in this sequence be a. Then the 18 terms are the 9 odd integers less than a and the 9 odd integers greater than a:
a-17, a-15, ..., a-1, a+1, ..., a+15, and a+17.
And the sum of the 18 terms is 18a.
If the sum of the 18 terms is a perfect 5th power of n, then
For that to be a perfect 5th power, the smallest possible value for a is
Then
So n is 6; and the smallest term in the sequence is a-17 = 432-17 = 415.
Then, finally, nx is 6*415 = 2490.
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