2015 AMC 8 Problems/Problem 18
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, is an arithmetic sequence with five terms, in which the first term is and the constant added is . Each row and each column in this array is an arithmetic sequence with five terms. The square in the center is labelled as shown. What is the value of ?
We begin filling in the table. The top row has a first term and a fifth term , so we have the common difference is . This means we can fill in the first row of the table:
The fifth row has a first term of and a fifth term of , so the common difference is . We can fill in the fifth row of the table as shown:
We must find the third term of the arithmetic sequence with a first term of and a fifth term of . The common difference of this sequence is , so the third term is .
The middle term of the first row is , since the middle number is just the average in an arithmetic sequence. Similarly, the middle of the bottom row is . Applying this again for the middle column, the answer is .
The value of is simply the average of the average values of both diagonals that contain . This is
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