2008 AMC 10B Problems/Problem 13
Contents
Problem
For each positive integer , the mean of the first terms of a sequence is . What is the term of the sequence?
Solution 1
Since the mean of the first terms is , the sum of the first terms is . Thus, the sum of the first terms is and the sum of the first terms is . Hence, the term of the sequence is
Note that is the sum of the first n odd integers.
Solution 2
Let be the terms of the sequence. We know , so we must have . The sum of consecutive odd numbers down to is a perfect square, if you don't believe me, try drawing squares with the sum, so , so the answer is .
Solution 3
Let the mean be
Note that this is also equal to n
1st term + nth term Now note that, from previous solutions, the first term is 1, hence the 2008th term is
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Solution 4 (Using Answer Choices)
From inspection, we see that the sum of the sequence is . We also notice that is the sum of the first odd integers. Because is the only odd integer, is the answer.
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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All AMC 10 Problems and Solutions |
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