Question

If three numbers \( a, b, c \) constitute both an (A)P and G.P, then

• A. \( a = b = c \)
• B. \( a = b + c \)
• C. \( ab = c \)
• D. \( a = b - c \)

Hint:

When solving for numbers that need to satisfy conditions for both an (A)P and G.P, use the properties of both sequences and solve the resulting system of equations.

Answer 1

The Correct Option is A

For three numbers \( a, b, c \) to form both an Arithmetic Progression ((A)P) and a Geometric Progression (G.P), they must satisfy the conditions for both sequences. 1. For (A)P: The condition for \( a, b, c \) to be in Arithmetic Progression is that the middle term \( b \) should be the average of the other two terms: \[ b = \frac{a + c}{2} \] 2. For G.P: The condition for \( a, b, c \) to be in Geometric Progression is that the square of the middle term \( b \) should be the product of the other two terms: \[ b^2 = ac \] Now, substituting \( b = \frac{a + c}{2} \) into the G.P condition: \[ \left( \frac{a + c}{2} \right)^2 = ac \] Simplifying the equation: \[ \frac{(a + c)^2}{4} = ac \] \[ (a + c)^2 = 4ac \] Expanding both sides: \[ a^2 + 2ac + c^2 = 4ac \] Rearranging terms: \[ a^2 + c^2 = 2ac \] This is the equation of a perfect square: \[ (a - c)^2 = 0 \] Therefore, \( a = c \). Since \( a = c \) and we already have \( b = \frac{a + c}{2} \), we conclude that: \[ a = b = c \] Thus, the answer is \( a = b = c \).