If a, 1, b are in arithmetic progression and 1, a, b are in geometric progression, then a and b are respectively equate to _______ (where a ? b).
-
A.
2, 4
-
B.
-2, 4
-
C.
4, 1
-
D.
-1, 2
Correct Answer:
@.
Explanation:
To determine the values of a and b, we use the definitions of arithmetic and geometric progressions.1. Arithmetic Progression (AP): For the sequence a, 1, b to be in AP, the common difference must be constant. This means:
1 - a = b - 1
a + b = 2 (Equation 1)
2. Geometric Progression (GP): For the sequence 1, a, b to be in GP, the common ratio must be constant. This means:
a / 1 = b / a
a^2 = b (Equation 2)
3. Substitution: Substitute Equation 2 into Equation 1:
a + a^2 = 2
a^2 + a - 2 = 0
4. Solving the Quadratic Equation: Factoring the equation gives:
(a + 2)(a - 1) = 0
This results in two possible values for a: -2 or 1.
5. Finding b: - If a = 1, then b = (1)^2 = 1. However, the problem states that a is not equal to b, so we discard this solution.
- If a = -2, then b = (-2)^2 = 4.Therefore, the values of a and b are -2 and 4, respectively.
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