`a`, `b`, `c`, `d` are in increasing `G.P.` If the `AM` between `a` and `b` is `6` and the `AM` between `c` and `d` is `54`, then the `AM` of `a` and `d` is

Options

A. `15`
B. `48`
C. `44`
D. `42`

Explanation

To solve the problem step by step, we will follow the information given and derive the required arithmetic mean (AM) between `a` and `d`.

### Step 1: Define the terms in the G.P.
Given that `a`, `b`, `c`, and `d` are in increasing geometric progression (G.P.), we can express them as:
- \( b = ar \)
- \( c = ar^2 \)
- \( d = ar^3 \)

where \( r \) is the common ratio.

### Step 2: Use the information about the AM between `a` and `b`
The arithmetic mean (AM) between `a` and `b` is given as 6:
\[ AM(a, b) = \frac{a + b}{2} = 6 \]
Substituting \( b = ar \):
\[ \frac{a + ar}{2} = 6 \]
Multiplying both sides by 2:
\[ a + ar = 12 \]
Factoring out \( a \):
\[ a(1 + r) = 12 \quad \text{(Equation 1)} \]

### Step 3: Use the information about the AM between `c` and `d`
The AM between `c` and `d` is given as 54:
\[ AM(c, d) = \frac{c + d}{2} = 54 \]
Substituting \( c = ar^2 \) and \( d = ar^3 \):
\[ \frac{ar^2 + ar^3}{2} = 54 \]
Multiplying both sides by 2:
\[ ar^2 + ar^3 = 108 \]
Factoring out \( ar^2 \):
\[ ar^2(1 + r) = 108 \quad \text{(Equation 2)} \]

### Step 4: Divide Equation 2 by Equation 1
Now we will divide Equation 2 by Equation 1:
\[ \frac{ar^2(1 + r)}{a(1 + r)} = \frac{108}{12} \]
Since \( 1 + r \) is common in both the numerator and denominator, we can cancel it out (assuming \( r \neq -1 \)):
\[ \frac{r^2}{1} = 9 \]
Thus, we have:
\[ r^2 = 9 \implies r = 3 \quad \text{(since \( r \) is positive)} \]

### Step 5: Substitute \( r \) back into Equation 1 to find \( a \)
Now substituting \( r = 3 \) back into Equation 1:
\[ a(1 + 3) = 12 \]
\[ a(4) = 12 \implies a = 3 \]

### Step 6: Calculate `d`
Now we can find \( d \):
\[ d = ar^3 = 3 \cdot 3^3 = 3 \cdot 27 = 81 \]

### Step 7: Find the AM between `a` and `d`
Finally, we find the AM between `a` and `d`:
\[ AM(a, d) = \frac{a + d}{2} = \frac{3 + 81}{2} = \frac{84}{2} = 42 \]

### Conclusion
The arithmetic mean of `a` and `d` is \( \boxed{42} \).