Difference between revisions of "2022 AMC 10B Problems/Problem 15"
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− | Suppose that the first number of the arithmetic sequence is <math>a</math>. We will try to compute the value of <math>S_{n}</math>. 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 | + | Suppose that the first number of the arithmetic sequence is <math>a</math>. We will try to compute the value of <math>S_{n}</math>. 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 |
~mathboy100 | ~mathboy100 |
Revision as of 01:29, 3 November 2024
Contents
[hide]Problem
Let be the sum of the first terms of an arithmetic sequence that has a common difference of . The quotient does not depend on . What is ?
Solution 1
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
~mathboy100
Solution 2
Let's say that our sequence is Then, since the value of n doesn't matter in the quotient , we can say that Simplifying, we get , from which Solving for , we get that .
Since the sum of the first odd numbers is , .
Solution 3 (Quick Insight)
Recall that the sum of the first odd numbers is .
Since , we have .
~numerophile
Video Solution (🚀 Solved in 4 min 🚀)
~Education, the Study of Everything
Video Solution By SpreadTheMathLove
https://www.youtube.com/watch?v=zHJJyMlH9DA
Video Solution by Interstigation
Video Solution by paixiao
https://www.youtube.com/watch?v=4bzuoKi2Tes
Video Solution by TheBeautyofMath
https://youtu.be/Mi2AxPhnRno?t=1299
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 |
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