Difference between revisions of "2021 AMC 10A Problems/Problem 4"
MRENTHUSIASM (talk | contribs) |
MRENTHUSIASM (talk | contribs) |
||
Line 5: | Line 5: | ||
==Solution== | ==Solution== | ||
− | Since <cmath>\text{Distance}=\text{Speed}\times\text{Time},</cmath> we seek the sum <cmath>5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,</cmath> in which there are 30 addends. The last addend is <math>5+7(30-1)=208.</math> Therefore, the requested sum is <cmath>5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\ | + | Since <cmath>\text{Distance}=\text{Speed}\times\text{Time},</cmath> we seek the sum <cmath>5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,</cmath> in which there are 30 addends. The last addend is <math>5+7(30-1)=208.</math> Therefore, the requested sum is <cmath>5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\textbf{(D)} ~3195}.</cmath> 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 <cmath>\frac{\text{First}+\text{Last}}{2}\cdot\text{Count}.</cmath> ~MRENTHUSIASM |
== Video Solution (Using Arithmetic Sequence) == | == Video Solution (Using Arithmetic Sequence) == |
Revision as of 22:43, 11 February 2021
Problem 4
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
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
Video Solution (Using Arithmetic Sequence)
~ pi_is_3.14