2021 AMC 10A Problem 4

Problem 4 of 25EasierAlgebra

A cart rolls down a hill, traveling

5

inches the first second and accelerating so that during each successive

1

-second time interval, it travels

7

inches more than during the previous

1

-second interval. The cart takes

30

seconds to reach the bottom of the hill. How far, in inches, does it travel?

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Solution

The distance travelled every second forms an arithmetic sequence:

5, 5 + 7, 5 + 2 \cdot 7, \ldots

The standard arithmetic-sequence sum formula is

n\cdot\dfrac{a_1+a_n}{2}.

We know the number of terms is

30

and the first term is

5.

The last term is

5 + 29 \cdot 7 = 208.

Plugging these values into the expression yields

30 \cdot \dfrac{5 + 208}{2} = 15 \cdot 213 = 3195.

Thus, D is the correct answer.