To solve the problem step by step, we will analyze the relationships between the terms given in the question.
### Step 1: Understanding the relationships
We have three sets of terms:
1. \( a, b, c \) are in Arithmetic Progression (A.P.)
2. \( b, c, d \) are in Geometric Progression (G.P.)
3. \( c, d, e \) are in Harmonic Progression (H.P.)
### Step 2: Expressing the conditions mathematically
1. Since \( a, b, c \) are in A.P., we can express this as:
\[
2b = a + c \quad \Rightarrow \quad b = \frac{a + c}{2}
\]
2. Since \( b, c, d \) are in G.P., we can express this as:
\[
c^2 = bd
\]
3. Since \( c, d, e \) are in H.P., we can express this as:
\[
d = \frac{2ce}{c + e}
\]
### Step 3: Substituting the expressions
Now, we will substitute the expression for \( b \) into the G.P. condition and the expression for \( d \) into the H.P. condition.
From the G.P. condition:
\[
c^2 = b \cdot d
\]
Substituting \( b \):
\[
c^2 = \left(\frac{a + c}{2}\right) d
\]
From the H.P. condition:
\[
d = \frac{2ce}{c + e}
\]
Substituting this into the G.P. condition:
\[
c^2 = \left(\frac{a + c}{2}\right) \left(\frac{2ce}{c + e}\right)
\]
### Step 4: Simplifying the equation
Now, simplifying the equation:
\[
c^2 = \frac{(a + c)ce}{c + e}
\]
Cross-multiplying gives:
\[
c^2(c + e) = (a + c)ce
\]
Expanding both sides:
\[
c^3 + c^2e = ace + c^2e
\]
Cancelling \( c^2e \) from both sides:
\[
c^3 = ace
\]
### Step 5: Concluding the relationships
From the equation \( c^3 = ace \), we can rearrange it to:
\[
c^2 = ae
\]
This indicates that \( a, c, e \) are in G.P. (since \( c^2 = ae \) is the condition for three numbers to be in G.P.).
### Final Answer
Thus, we conclude that \( a, c, e \) are in Geometric Progression (G.P.).
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