2021 AMC 10A Spring Problem 4

Question

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?

215

360

2992

3195

3242

The distances traveled each second form an arithmetic sequence

Find the

30

th term, then use the average of the first and last terms

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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.

Problem 4 in other years