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Latest revision as of 10:28, 13 January 2024
Contents
Problem
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
?
Solutions
Solution 1
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
.
Solution 2
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
.
Solution 3
The value of is simply the average of the average values of both diagonals that contain
. This is
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/tKsYSBdeVuw?t=1258
Video Solution
~savannahsolver
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.