If a,b,c, are in A.P., b,c,d are in G.P. and c,d,e, are in H.P., then a,c,e are in

Options

A. (1) No particular order
B. (2) A.P
C. (3) G.P
D. (4) H.P

Answer

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

### Step 1: Understanding the given conditions
We have three sets of numbers:
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: Using the A.P. condition
For \( a, b, c \) to be in A.P., we can use the formula for the arithmetic mean:
\[ b = \frac{a + c}{2} \]
This can be rearranged to give us:
\[ a + c = 2b \quad \text{(Equation 1)} \]

### Step 3: Using the G.P. condition
For \( b, c, d \) to be in G.P., we can use the formula for the geometric mean:
\[ c^2 = b \cdot d \]
From this, we can express \( d \) in terms of \( b \) and \( c \):
\[ d = \frac{c^2}{b} \quad \text{(Equation 2)} \]

### Step 4: Using the H.P. condition
For \( c, d, e \) to be in H.P., we can use the formula for the harmonic mean:
\[ \frac{2}{d} = \frac{1}{c} + \frac{1}{e} \]
Rearranging this gives:
\[ \frac{2}{d} = \frac{c + e}{ce} \]
Cross-multiplying, we get:
\[ 2ce = d(c + e) \quad \text{(Equation 3)} \]

### Step 5: Substitute \( d \) from Equation 2 into Equation 3
Substituting \( d = \frac{c^2}{b} \) into Equation 3:
\[ 2ce = \left(\frac{c^2}{b}\right)(c + e) \]
Multiplying both sides by \( b \):
\[ 2bce = c^2(c + e) \]
Expanding the right-hand side:
\[ 2bce = c^3 + c^2e \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 2bce - c^2e = c^3 \]
Factoring out \( e \) from the left side:
\[ e(2bc - c^2) = c^3 \]
Thus, we have:
\[ e = \frac{c^3}{2bc - c^2} \quad \text{(Equation 4)} \]

### Step 7: Finding the relationship between \( a, c, e \)
From Equation 1, we know \( a + c = 2b \). We can express \( b \) in terms of \( a \) and \( c \):
\[ b = \frac{a + c}{2} \]
Substituting this expression for \( b \) into our equation for \( e \):
\[ e = \frac{c^3}{2 \left(\frac{a+c}{2}\right)c - c^2} \]
This simplifies to:
\[ e = \frac{c^3}{(a + c)c - c^2} = \frac{c^3}{ac} \]
Thus, we have:
\[ e = \frac{c^2}{a} \]

### Step 8: Conclusion
Now we have \( a, c, e \) in the form:
\[ a = c^2/e \]
This indicates that \( a, c, e \) are in Geometric Progression (G.P.) since the relationship satisfies the condition for G.P.

### Final Answer
Thus, \( a, c, e \) are in **Geometric Progression (G.P.)**.

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