If a,b, and c are in G.P then a+b,2b and b+ c are in

Options

A. A.P
B. G.P
C. H.P
D. none of these

Explanation

To determine whether \( a + b, 2b, \) and \( b + c \) are in an arithmetic progression (AP) given that \( a, b, \) and \( c \) are in geometric progression (G.P.), we can follow these steps:

### Step 1: Define the terms in G.P.
Since \( a, b, c \) are in G.P., we can express them in terms of a common ratio \( r \):
- Let \( a = A \)
- Let \( b = Ar \)
- Let \( c = Ar^2 \)

### Step 2: Write the terms \( a + b, 2b, b + c \)
Now we can express the terms \( a + b, 2b, \) and \( b + c \):
- \( a + b = A + Ar = A(1 + r) \)
- \( 2b = 2(Ar) = 2Ar \)
- \( b + c = Ar + Ar^2 = Ar(1 + r) \)

### Step 3: Check if these terms are in AP
For three terms \( x, y, z \) to be in AP, the condition is:
\[ 2y = x + z \]
Substituting our expressions:
\[ 2(2Ar) = A(1 + r) + Ar(1 + r) \]
This simplifies to:
\[ 4Ar = A(1 + r) + Ar(1 + r) \]

### Step 4: Simplify the right-hand side
Now, let's simplify the right-hand side:
\[ A(1 + r) + Ar(1 + r) = A(1 + r) + Ar(1 + r) = A(1 + r) + A(1 + r)r = A(1 + r)(1 + r) = A(1 + r)^2 \]

### Step 5: Set the equation
Now we have:
\[ 4Ar = A(1 + r)^2 \]
Assuming \( A \neq 0 \) (since if \( A = 0 \), all terms are zero), we can divide both sides by \( A \):
\[ 4r = (1 + r)^2 \]

### Step 6: Expand and rearrange the equation
Expanding the right-hand side:
\[ 4r = 1 + 2r + r^2 \]
Rearranging gives us:
\[ r^2 - 2r + 1 = 0 \]
This can be factored as:
\[ (r - 1)^2 = 0 \]
Thus, \( r = 1 \).

### Step 7: Conclusion
Since \( r = 1 \), it confirms that \( a, b, c \) are equal, and therefore \( a + b, 2b, b + c \) are indeed in AP.

### Final Answer
Thus, \( a + b, 2b, b + c \) are in Arithmetic Progression (AP).

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