2008 AMC 10B Problems/Problem 13
Contents
[hide]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 (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.
Solution 3 (Basically Solution Two Just More Rigorous)
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 .
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.