Question

a high school basketball game between the raiders and wildcats was tied at the end of the first quarter. the number of points scored by the raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the wildcats in each of the four quarters formed an increasing arithmetic sequence. at the end of the fourth quarter, the raiders had won by one point. neither team scored more than $100$ points. what was the total number of points scored by the two teams in the first half? $\textbf{(a)}\ 30 \qquad \textbf{(b)}\ 31 \qquad \textbf{(c)}\ 32 \qquad \textbf{(d)}\ 33 \qquad \textbf{(e)}\ 34$

Answer 1

The total number of points scored by the two teams in the first half is 32.

Step-by-step explanation:

Let's say the number of points scored by the Raiders in the first quarter is a. Since the number of points forms an increasing geometric sequence, the points scored in the other quarters can be represented as ar, ar^2, and ar^3. Similarly, let's say the number of points scored by the Wildcats in the first quarter is b. Since the number of points forms an increasing arithmetic sequence, the points scored in the other quarters can be represented as b+d, b+2d, and b+3d.

According to the problem, at the end of the fourth quarter, the Raiders had won by one point. This means the sum of the points scored in the four quarters by the Raiders is one more than the sum of the points scored by the Wildcats. We can set up the equation:

a + ar + ar^2 + ar^3 = b + b + d + b + 2d + b + 3d + 1

Since neither team scored more than 100 points, we can examine the values of a and b between 1 and 100 that satisfy the equation. By doing so, we find that the only possible value for the sum of the two teams' scores in the first half is 32.

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