Let a,b,c,d, e ar non-zero and distinct positive real numbers. If a,b, c are In a,b,c are in A.B, b,c, dare in G.P. and c,d e are in H.P, the a,c,e are in :

Options

A. A.P.
B. G.P.
C. H.P.
D. Nothing can be said

Answer

To solve the problem step by step, we will analyze the given conditions and derive the necessary relationships.

**Step 1: Understand the given conditions.**
We have three sequences:
1. \( a, b, c \) are in Arithmetic Progression (AP).
2. \( b, c, d \) are in Geometric Progression (GP).
3. \( c, d, e \) are in Harmonic Progression (HP).

**Step 2: Use the definition of AP for \( a, b, c \).**
Since \( a, b, c \) are in AP, we can express this as:
\[ b = \frac{a + c}{2} \tag{1} \]

**Step 3: Use the definition of GP for \( b, c, d \).**
Since \( b, c, d \) are in GP, we can express this as:
\[ c^2 = bd \tag{2} \]

**Step 4: Use the definition of HP for \( c, d, e \).**
Since \( c, d, e \) are in HP, we can express this as:
\[ d = \frac{2ce}{c + e} \tag{3} \]

**Step 5: Substitute equations (1) and (3) into equation (2).**
From equation (1), we have \( b = \frac{a + c}{2} \). Substitute this into equation (2):
\[ c^2 = \left(\frac{a + c}{2}\right) d \]
Now, substitute \( d \) from equation (3):
\[ c^2 = \left(\frac{a + c}{2}\right) \left(\frac{2ce}{c + e}\right) \]
This simplifies to:
\[ c^2 = \frac{(a + c) ce}{c + e} \]

**Step 6: Cross-multiply to eliminate the fraction.**
Cross-multiplying gives:
\[ c^2(c + e) = (a + c) ce \]
Expanding both sides:
\[ c^3 + c^2 e = ace + c e^2 \]

**Step 7: Rearranging the equation.**
Rearranging gives:
\[ c^3 - ace + c^2 e - c e^2 = 0 \]

**Step 8: Factor out common terms.**
We can factor out \( c \):
\[ c(c^2 - ae + ce - e^2) = 0 \]
Since \( c \) is non-zero, we can ignore this factor:
\[ c^2 - ae + ce - e^2 = 0 \]

**Step 9: Rearranging to find the relationship between \( a, c, e \).**
This leads us to:
\[ c^2 = ae \]
This is the condition for \( a, c, e \) to be in Geometric Progression (GP).

**Conclusion:**
Since we have derived that \( c^2 = ae \), we conclude that \( a, c, e \) are in Geometric Progression (GP).

**Final Answer:**
\( a, c, e \) are in GP.

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