`a`, `b`, `c` are positive integers formaing an incresing `G.P.` and `b-a` is a perfect cube and `log_(6)a+log_(6)b+log_(6)c=6`, then `a+b+c=`

Options

A. `100`
B. `111`
C. `122`
D. `189`

Explanation

To solve the problem step by step, we need to follow the given conditions and properties of logarithms and geometric progressions.

### Step 1: Understand the properties of G.P.
Given that \( a, b, c \) are in an increasing geometric progression (G.P.), we can express the relationship between them as:
\[ b^2 = ac \]

### Step 2: Use the logarithmic equation
We are given the equation:
\[ \log_6 a + \log_6 b + \log_6 c = 6 \]
Using the property of logarithms that states \( \log_a b + \log_a c = \log_a (bc) \), we can rewrite this as:
\[ \log_6 (abc) = 6 \]
This implies:
\[ abc = 6^6 \]

### Step 3: Calculate \( 6^6 \)
Calculating \( 6^6 \):
\[ 6^6 = 46656 \]
Thus, we have:
\[ abc = 46656 \]

### Step 4: Use the condition \( b - a \) is a perfect cube
Let \( b - a = n^3 \) for some positive integer \( n \). Therefore, we can express \( b \) as:
\[ b = a + n^3 \]

### Step 5: Substitute \( b \) in the G.P. relationship
Substituting \( b \) into the G.P. relationship \( b^2 = ac \):
\[ (a + n^3)^2 = ac \]
Expanding this gives:
\[ a^2 + 2an^3 + n^6 = ac \]

### Step 6: Express \( c \) in terms of \( a \) and \( n \)
From \( abc = 46656 \), we can express \( c \) as:
\[ c = \frac{46656}{ab} \]

### Step 7: Substitute \( c \) into the G.P. equation
Substituting \( c \) into the G.P. equation:
\[ (a + n^3)^2 = a \left( \frac{46656}{ab} \right) \]
This simplifies to:
\[ (a + n^3)^2 = \frac{46656}{b} \]

### Step 8: Find suitable values for \( a \) and \( b \)
To satisfy both the G.P. condition and the perfect cube condition, let’s try some values. We can start with \( a = 9 \) and \( b = 36 \):
\[ b - a = 36 - 9 = 27 = 3^3 \quad (\text{which is a perfect cube}) \]

### Step 9: Calculate \( c \)
Now we can find \( c \):
\[ abc = 46656 \implies 9 \cdot 36 \cdot c = 46656 \]
Calculating \( 9 \cdot 36 = 324 \):
\[ 324c = 46656 \implies c = \frac{46656}{324} = 144 \]

### Step 10: Calculate \( a + b + c \)
Now we can find the sum:
\[ a + b + c = 9 + 36 + 144 = 189 \]

### Final Answer
Thus, the value of \( a + b + c \) is:
\[ \boxed{189} \]