Difference between revisions of "2021 AMC 10A Problems/Problem 4"
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From the <math>30</math>-term sum <cmath>5+12+19+26+\cdots</cmath> in the previous solution, taking modulo <math>10</math> gives <cmath>5+12+19+26+\cdots \equiv 3(0+1+2+3+4+5+6+7+8+9) = 3(45)\equiv5 \pmod{10}.</cmath> The only answer choices that are <math>5\mod{10}</math> are <math>\textbf{(A)}</math> and <math>\textbf{(D)}.</math> By a quick estimate, <math>\textbf{(A)}</math> is too small, leaving us with <math>\boxed{\textbf{(D)} ~3195}.</math> ~MRENTHUSIASM | From the <math>30</math>-term sum <cmath>5+12+19+26+\cdots</cmath> in the previous solution, taking modulo <math>10</math> gives <cmath>5+12+19+26+\cdots \equiv 3(0+1+2+3+4+5+6+7+8+9) = 3(45)\equiv5 \pmod{10}.</cmath> The only answer choices that are <math>5\mod{10}</math> are <math>\textbf{(A)}</math> and <math>\textbf{(D)}.</math> By a quick estimate, <math>\textbf{(A)}</math> is too small, leaving us with <math>\boxed{\textbf{(D)} ~3195}.</math> ~MRENTHUSIASM | ||
− | == Video Solution (Arithmetic Sequence but in a different way) | + | == Video Solution == |
+ | (Arithmetic Sequence but in a different way) | ||
https://www.youtube.com/watch?v=sJa7uB-UoLc&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=4 | https://www.youtube.com/watch?v=sJa7uB-UoLc&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=4 | ||
Revision as of 18:09, 14 February 2021
Contents
Problem
A cart rolls down a hill, travelling inches the first second and accelerating so that during each successive
-second time interval, it travels
inches more than during the previous
-second interval. The cart takes
seconds to reach the bottom of the hill. How far, in inches, does it travel?
Solution 1
Since we seek the sum
in which there are 30 addends. The last addend is
Therefore, the requested sum is
Recall that to find the sum of an arithmetic series, we take the average of the first and last terms, then multiply by the number of terms, namely
~MRENTHUSIASM
Solution 2 (Answer Choices and Modular Arithmetic)
From the -term sum
in the previous solution, taking modulo
gives
The only answer choices that are
are
and
By a quick estimate,
is too small, leaving us with
~MRENTHUSIASM
Video Solution
(Arithmetic Sequence but in a different way) https://www.youtube.com/watch?v=sJa7uB-UoLc&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=4
~ North America Math Contest Go Go Go
Video Solution (Using Arithmetic Sequence)
~ pi_is_3.14
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.