2011 AMC 10A Problems/Problem 4
Contents
Problem
Let X and Y be the following sums of arithmetic sequences:
What is the value of
Solution 1
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:
From here it is obvious that .
Note
Another way to see this is to let the sum So, the sequences become
Like before, the difference between the two sequences is
Solution 2
We see that every number in Y's sequence is two more than every corresponding number in X's sequence. Since there are 46 numbers in each sequence, the difference must be
Solution 3

Solution 4
In an actual contest, this would probably take too much time but is nevertheless a solution. The general formula for computing sums of any arithmetic sequence where is the number of terms, is the first term and is the last term is . If one uses that formula for both sequences, they will get as the sum for and as the sum for . Subtracting from , one will get the answer .  danfan
Video Solution
~savannahsolver
See Also
2011 AMC 10A (Problems • Answer Key • Resources)  
Preceded by Problem 3 
Followed by Problem 5  
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All AMC 10 Problems and Solutions 
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