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.