Difference between revisions of "2022 AMC 10B Problems/Problem 15"
MRENTHUSIASM (talk | contribs) m |
MRENTHUSIASM (talk | contribs) m |
||
Line 11: | Line 11: | ||
~mathboy100 | ~mathboy100 | ||
− | ==Solution 2 (Quick Insight)== | + | ==Solution 2== |
+ | Let's say that our sequence is <cmath>a, a+2, a+4, a+6, a+8, a+10, \ldots.</cmath> | ||
+ | Then, since the value of n doesn't matter in the quotient <math>\frac{S_{3n}}{S_n}</math>, we can say that | ||
+ | <cmath>\frac{S_{3}}{S_1} = \frac{S_{6}}{S_2}.</cmath> | ||
+ | Simplifying, we get <math>\frac{3a+6}{a}=\frac{6a+30}{2a+2}</math>, from which <cmath>\frac{a+2}{a}=\frac{a+5}{a+1}.</cmath> | ||
+ | Solving for <math>a</math>, we get that <math>a=1</math>. Now, we proceed similar to Solution 1 and get that <math>S_{20} = 20^2 = \boxed{\textbf{(D) } 400}</math>. | ||
+ | |||
+ | ==Solution 3 (Quick Insight)== | ||
Recall that the sum of the first <math>n</math> odd numbers is <math>n^2</math>. | Recall that the sum of the first <math>n</math> odd numbers is <math>n^2</math>. | ||
Line 18: | Line 25: | ||
~numerophile | ~numerophile | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
==Video Solution 1== | ==Video Solution 1== |
Revision as of 06:54, 28 November 2022
Contents
[hide]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 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 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, we have .
~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 . Now, we proceed similar to Solution 1 and get that .
Solution 3 (Quick Insight)
Recall that the sum of the first odd numbers is .
Since , we have .
~numerophile
Video Solution 1
~Education, the Study of Everything
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.