If (a - b) = 3 , (b - c) = 5 and ( c - a) = 1 , then the value of `(a^(3) + b^(3) + c^(3) - 3abc)/(a + b + c)` is

Options

A. `17.5`
B. `20.5`
C. `10.5`
D. `15.5`

Answer

To solve the problem, we will use the given equations and the formula for \( \frac{a^3 + b^3 + c^3 - 3abc}{a + b + c} \).

### Step 1: Write down the given equations
We have:
1. \( a - b = 3 \) (Equation 1)
2. \( b - c = 5 \) (Equation 2)
3. \( c - a = 1 \) (Equation 3)

### Step 2: Express \( a \), \( b \), and \( c \) in terms of one variable
From Equation 1:
\[ a = b + 3 \]

From Equation 2:
\[ b = c + 5 \]
Substituting for \( b \) in terms of \( c \):
\[ a = (c + 5) + 3 = c + 8 \]

From Equation 3:
\[ c = a - 1 \]
Substituting \( a \) in terms of \( c \):
\[ c = (c + 8) - 1 \]
This simplifies to:
\[ c = c + 7 \]
This indicates that we need to solve for \( c \) in terms of \( a \).

### Step 3: Solve the equations
Let’s express everything in terms of \( c \):
From \( b = c + 5 \):
\[ b = c + 5 \]

From \( a = c + 8 \):
\[ a = c + 8 \]

Now, we can substitute these values back into the equations to find \( a \), \( b \), and \( c \).

### Step 4: Calculate \( a + b + c \)
Now we can find \( a + b + c \):
\[ a + b + c = (c + 8) + (c + 5) + c = 3c + 13 \]

### Step 5: Calculate \( a^3 + b^3 + c^3 - 3abc \)
Using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c) \left( (a - b)^2 + (b - c)^2 + (c - a)^2 \right) \]
We already have \( a + b + c = 3c + 13 \).

Next, we need to calculate \( (a - b)^2 + (b - c)^2 + (c - a)^2 \):
- \( (a - b)^2 = (3)^2 = 9 \)
- \( (b - c)^2 = (5)^2 = 25 \)
- \( (c - a)^2 = (-1)^2 = 1 \)

Adding these:
\[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 9 + 25 + 1 = 35 \]

### Step 6: Substitute into the identity
Now substituting into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (3c + 13)(35) \]

### Step 7: Calculate the final expression
Now we can find:
\[ \frac{a^3 + b^3 + c^3 - 3abc}{a + b + c} = \frac{(3c + 13)(35)}{3c + 13} \]

This simplifies to:
\[ 35 \]

### Final Answer
Thus, the value of \( \frac{a^3 + b^3 + c^3 - 3abc}{a + b + c} \) is \( 35 \).

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