2021 AMC 10A Problems/Problem 4
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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 ![\[\text{Distance}=\text{Speed}\times\text{Time},\]](http://latex.artofproblemsolving.com/6/0/6/606216cd23e3b2f30aad3c379f14bbe7df7522e0.png)
we seek the sum ![\[5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,\]](http://latex.artofproblemsolving.com/8/c/7/8c79e9f6d74d79045a0538ba9ba95d45a5f6d918.png)
in which there are 30 addends. The last addend is 
Therefore, the requested sum is ![\[5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\text{(D) } 3195}.\]](http://latex.artofproblemsolving.com/a/a/d/aadb9bbead55745590da3bf3ff43a13a2b80085f.png)
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 ![\[\frac{\text{First}+\text{Last}}{2}\cdot\text{Count}.\]](http://latex.artofproblemsolving.com/f/e/6/fe6482f623264044d4444569428eb2cc7d8c0d9f.png)
Video Solution (Using Arithmetic Sequence)
~ pi_is_3.14
