Question

A high school basketball game between the Raiders and the 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

Answer 1

Answer: 5 and 14.

Explanation:

We know that the Raiders and Wildcats both scored the same number of points in the first quarter so let a,a+d,a+2d,a+3d be the quarterly scores for the Wildcats. The sum of the Raiders scores is a(1+r+r^{2}+r^{3}) and the sum of the Wildcats scores is 4a+6d. Now we can narrow our search for the values of a,d, and r. Because points are always measured in positive integers, we can conclude that a and d are positive integers. We can also conclude that $r$ is a positive integer by writing down the equation:

a(1+r+r^{2}+r^{3})=4a+6d+1

Now we can start trying out some values of r. We try r=2, which gives

15a=4a+6d+1

11a=6d+1

We need the smallest multiple of 11 (to satisfy the <100 condition) that is 1 (mod 6). We see that this is 55, and therefore a=5 and d=9.

So the Raiders' first two scores were 5 and 10 and the Wildcats' first two scores were 5 and 14.