Question

In an 8-week course, the teacher conducts a test every week, and the scores are in the range of 1–4. There are only two students enrolled in the course, \(R\) and \(S\). The following conditions are given:

1. \(R\) and \(S\) scored the same on the first test.
2. From the second test onwards, \(R\) consistently scored the same (a non-zero score).
3. The total of the first three test scores of \(R\) equals the total of the first two test scores of \(S\).
4. From the fifth test onwards, \(S\) scored the same as \(R\).
5. The scores of \(S\) from the first two tests and all the remaining tests follow a geometric progression.

Hint:

In problems with geometric progressions, identify the common ratio r and use boundary conditions to limit possible values. Check all constraints step-by-step for consistency.

Answer 1

Let the scores of \(R\) and \(S\) for the tests be as follows:

• Scores of \(R\): \(R_1, R_2, R_3, R_4, R_5, R_6, R_7, R_8\)
• Scores of \(S\): \(S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8\)

Step 1: Analyze the conditions

1. From condition 1: \(R_1 = S_1\)
2. From condition 2: \(R_2 = R_3 = R_4 = R_5 = R_6 = R_7 = R_8 = k\), where \(k\) is a constant score.
3. From condition 3: \(R_1 + R_2 + R_3 = S_1 + S_2\)
4. From condition 4: \(S_5 = S_6 = S_7 = S_8 = k\)
5. From condition 5: The scores of \(S\) are in a geometric progression. Let \(S_1 = a\) and the common ratio of the geometric progression be \(r\):
   • \(S_2 = ar\), \(S_3 = ar^2\), \(S_4 = ar^3\), \(S_5 = k\)

Step 2: Solve the equations

1. From condition 3:
2. From condition 5, the geometric progression stops at \(S_5 = k\). For this to hold:

Step 3: Determine the values of \(a\) and \(k\)

Since scores range from 1 to 4, we test values of \(k\) and \(a\) to satisfy all conditions. Let \(k = 2\):

\[ a = 2k = 4 \]

Substitute \(a = 4\) and \(k = 2\) into the sequence:

• \(S_1 = 4, S_2 = ar = 4 \times \frac{2k}{a} = 4 \times \frac{4}{4} = 2\)
• \(S_3 = ar^2 = 4, S_4 = ar^3 = 2, S_5 = k = 2\)

Final Scores:

• \(R = [4, 2, 2, 2, 2, 2, 2, 2]\)
• \(S = [4, 2, 4, 2, 2, 2, 2, 2]\)

Verification:

1. \(R_1 + R_2 + R_3 = 4 + 2 + 2 = 8\), \(S_1 + S_2 = 4 + 4 = 8\). Condition satisfied.
2. \(S\) forms a geometric progression for \(S_1, S_2, S_3, S_4\). Condition satisfied.
3. From the fifth test onwards, \(S_5 = R_5 = 2\). Condition satisfied.