`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 start with the original equation:
\[ 5^{(x - 1)} + 5^x + 5^{(x + 1)} = 775 \]
We can express \( 5^{(x - 1)} \) and \( 5^{(x + 1)} \) in terms of \( 5^x \):
\[ 5^{(x - 1)} = \frac{5^x}{5} \quad \text{and} \quad 5^{(x + 1)} = 5 \cdot 5^x \]
Substituting these into the equation gives:
\[ \frac{5^x}{5} + 5^x + 5 \cdot 5^x = 775 \]

### Step 2: Combine like terms
Now, we can combine the terms on the left side:
\[ \frac{5^x}{5} + 5^x + 5 \cdot 5^x = \frac{5^x}{5} + 1 \cdot 5^x + 5 \cdot 5^x = \left(\frac{1}{5} + 1 + 5\right)5^x \]
Calculating the coefficients:
\[ \frac{1}{5} + 1 + 5 = \frac{1}{5} + \frac{5}{5} + \frac{25}{5} = \frac{31}{5} \]
Thus, we rewrite the equation as:
\[ \frac{31}{5} \cdot 5^x = 775 \]

### Step 3: Clear the fraction
To eliminate the fraction, 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: Divide both sides by 31
Now, divide both sides by 31 to isolate \( 5^x \):
\[ 5^x = \frac{3875}{31} \]
Calculating \( \frac{3875}{31} \):
\[ \frac{3875}{31} = 125 \]
Thus, we have:
\[ 5^x = 125 \]

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

### Final Answer
The value of \( x \) is:
\[ \boxed{3} \]