Question

\( a_1, a_2, a_3, \dots, a_n \) are in A.P. and sum of first 10 terms is 160. \( g_1, g_2, g_3, \dots, g_n \) are in G.P. where \( g_1 + g_2 = 8 \). If the first term of A.P. is equal to common ratio of G.P. and first term of G.P. is equal to common difference of A.P., then sum of all possible values of \( g_1 \) is equal to:

• A. \( \frac{34}{9} \)
• B. \( \frac{28}{9} \)
• C. \( \frac{23}{3} \)
• D. \( \frac{28}{5} \)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We represent the variables using the given definitions: - A.P.: first term \( a_1 \), common difference \( d \). - G.P.: first term \( g_1 \), common ratio \( r \). Given: \( a_1 = r \) and \( g_1 = d \). We establish a system of equations.

Step 2: Key Formula or Approach:
1. \( S_{10} = \frac{10}{2} [2a_1 + 9d] = 160 \implies 2r + 9g_1 = 32 \). 2. \( g_1 + g_1 r = 8 \implies g_1(1 + r) = 8 \).

Step 3: Detailed Explanation:
1. From eq (1): \( r = \frac{32 - 9g_1}{2} \). 2. Substitute into eq (2): \[ g_1 \left( 1 + \frac{32 - 9g_1}{2} \right) = 8 \] \[ g_1 \left( \frac{2 + 32 - 9g_1}{2} \right) = 8 \implies g_1(34 - 9g_1) = 16 \] 3. Form the quadratic equation: \[ 9g_1^2 - 34g_1 + 16 = 0 \] 4. Possible values of \( g_1 \) are the roots of this equation. 5. Sum of roots (sum of possible values of \( g_1 \)): \[ \text{Sum} = -\frac{b}{a} = \frac{34}{9} \] \textit{(Note: If one root leads to an invalid G.P. or A.P., it is excluded. Checking roots: \( (9g_1 - 4)(g_1 - 4) = 0 \). Roots are 4 and 4/9. If \( g_1 = 4 \), \( r = (32-36)/2 = -2 \). If \( g_1 = 4/9 \), \( r = (32-4)/2 = 14 \). Both are valid.)}

Step 4: Final Answer:
Sum of possible values of \( g_1 \) is \( \frac{34}{9} \).