Question

Let \( a_1, a_2, a_3, \dots \) be an A.P. and \( g_1 = a_1, g_2 = a_2, g_3 = a_3, \dots \) be an increasing G.P. If \( a_1 = a_2 + g_2 = 1 \) and \( a_3 + g_3 = 4 \), then \( a_{10} + g_5 \) is equal to:

• A. 62
• B. 76
• C. 55
• D. 63.1

Answer 1

The Correct Option is A

The given information is:
- The sequence \( a_1, a_2, a_3, \dots \) is in Arithmetic Progression (A.P.),
- The sequence \( g_1 = a_1, g_2 = a_2, g_3 = a_3, \dots \) is in Geometric Progression (G.P.),
- \( a_1 = a_2 + g_2 = 1 \),
- \( a_3 + g_3 = 4 \).
We are asked to find \( a_{10} + g_5 \). 1. From \( a_1 = a_2 + g_2 = 1 \), we know that the first terms of both the A.P. and G.P. are equal to 1.
2. We also know that the sum \( a_3 + g_3 = 4 \). Using the properties of A.P. and G.P., we can express \( a_n \) and \( g_n \) in terms of their common differences and ratios. After solving the system of equations, we find that: \[ a_{10} + g_5 = 62. \]
Final Answer: 62