2022 AMC 10B Problems/Problem 15
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Problem
Let be the sum of the first
term of an arithmetic sequence that has a common difference of
. The quotient
does not depend on
. What is
?
Solution
Suppose that the first number of the arithmetic sequence is . We will try to compute the value of
. First, note that the sum of an arithmetic sequence is equal to the number of terms multiplied by the median of the sequence. The median of this sequence is equal to
. Thus, the value of
is
. Then,
Of course, for this value to be constant,
must be
for all values of
, and thus
. Finally, the value of
is