`5^(x-1) + 5^x + 5^(x+1) = 775` then the value of x for every positive integer x, is :

Options

A. 1
B. 3
C. 2
D. can't be determined

Answer

To solve the equation \( 5^{(x-1)} + 5^x + 5^{(x+1)} = 775 \), we can follow these steps:

### Step 1: Rewrite the equation
We can express all terms with a common base of \( 5^x \):
\[ 5^{(x-1)} + 5^x + 5^{(x+1)} = 775 \]
This can be rewritten as:
\[ \frac{5^x}{5} + 5^x + 5 \cdot 5^x = 775 \]
This simplifies to:
\[ \frac{5^x}{5} + 5^x + 5^{x+1} = 775 \]

### Step 2: Factor out \( 5^x \)
Now, we can factor out \( 5^x \) from the left side:
\[ 5^x \left(\frac{1}{5} + 1 + 5\right) = 775 \]
Calculating the expression inside the parentheses:
\[ \frac{1}{5} + 1 + 5 = \frac{1}{5} + \frac{5}{5} + \frac{25}{5} = \frac{1 + 5 + 25}{5} = \frac{31}{5} \]
So, we have:
\[ 5^x \cdot \frac{31}{5} = 775 \]

### Step 3: Simplify the equation
Multiply both sides by \( 5 \):
\[ 31 \cdot 5^x = 775 \cdot 5 \]
Calculating \( 775 \cdot 5 \):
\[ 775 \cdot 5 = 3875 \]
So, we have:
\[ 31 \cdot 5^x = 3875 \]

### Step 4: Solve for \( 5^x \)
Now, divide both sides by \( 31 \):
\[ 5^x = \frac{3875}{31} \]
Calculating \( \frac{3875}{31} \):
\[ 3875 \div 31 = 125 \]
Thus:
\[ 5^x = 125 \]

### Step 5: Solve for \( x \)
Since \( 125 = 5^3 \), we can equate the exponents:
\[ x = 3 \]

### Conclusion
The value of \( x \) for every positive integer \( x \) is:
\[ \boxed{3} \]