8-Week Test Scores Progression

Let $x$ be the score of both in Test 1. Let $y \in \{1, 2, 3, 4\}$ be Ravi's score in Tests 2-8. Ravi's total in first 3 tests = $x + 2y$. Sumana's total in first 2 tests = $x + 2y \implies$ Sumana scored $2y$ in Test 2. Sumana's scores in 8 tests: $x, 2y, a, b, y, y, y, y$. Sumana's total in 8 tests = $x + 2y + a + b + 4y = x + a + b + 6y$. GP terms for Sumana: $T_1 = x$, $T_2 = x + 2y$, $T_3 = x + a + b + 6y$. Since $x \in \{1, 2, 3, 4\}$, we test possible GPs: Case: $x=2, y=1, a+b=0 \implies T_1=2, T_2=4, T_3=8$, where Sumana scored $x=2$ in Test 1. Hence, option B ("Sumana scored 2 marks in the first test") CAN be true.

If $x = 4$, then $T_1 = 4$. In the GP $(4, 8, 16)$ with $y = 2$ and $a + b = 0$, both $a, b \ge 0 \implies a = 0$ and $b = 0$. Sumana's score in the third test is $a = 0$.

If $y = 1$, possible GP cases are: Case 1: $x = 1, y = 1, a+b = 2 \implies T_1 = 1, T_2 = 3, T_3 = 9$. Case 2: $x = 2, y = 1, a+b = 0 \implies T_1 = 2, T_2 = 4, T_3 = 8$. Total marks for Sumana in Case 1 = 9. Total marks for Sumana in Case 2 = 8. Maximum possible value is 9.