The value of `(5.(25)^(n+1) + 25.(5)^(2n-1))/(25.(5)^(2n) -105(25)^(n-1))` is :

Options

A. 0
B. 1
C. `6(1/4)`
D. `5(1/4)`

Answer

To solve the expression \((5 \cdot (25)^{(n+1)} + 25 \cdot (5)^{(2n-1)}) / (25 \cdot (5)^{(2n)} - 105 \cdot (25)^{(n-1)})\), we will follow these steps:

### Step 1: Rewrite 25 in terms of 5
We know that \(25 = 5^2\). Therefore, we can rewrite the expression:
\[ (5 \cdot (5^2)^{(n+1)} + (5^2) \cdot (5)^{(2n-1)}) / ((5^2) \cdot (5)^{(2n)} - 105 \cdot (5^2)^{(n-1)}) \]

### Step 2: Simplify the powers
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the expression:
\[ (5 \cdot 5^{2(n+1)} + 5^2 \cdot 5^{(2n-1)}) / (5^2 \cdot 5^{2n} - 105 \cdot 5^{2(n-1)}) \]

### Step 3: Combine the exponents
Now we can combine the exponents in the numerator:
\[ = (5^{1 + 2(n+1)} + 5^{2 + (2n-1)}) / (5^{2 + 2n} - 105 \cdot 5^{2(n-1)}) \]
This simplifies to:
\[ = (5^{2n + 3} + 5^{2n + 1}) / (5^{2n + 2} - 105 \cdot 5^{2n - 2}) \]

### Step 4: Factor out common terms
In the numerator, we can factor out \(5^{2n + 1}\):
\[ = 5^{2n + 1}(5^2 + 1) / (5^{2n + 2} - 105 \cdot 5^{2n - 2}) \]
In the denominator, we can factor out \(5^{2n - 2}\):
\[ = 5^{2n - 2}(5^4 - 105) \]

### Step 5: Substitute and simplify
Now substituting back, we have:
\[ = \frac{5^{2n + 1} \cdot 26}{5^{2n - 2}(625 - 105)} \]
This simplifies to:
\[ = \frac{5^{2n + 1} \cdot 26}{5^{2n - 2} \cdot 520} \]

### Step 6: Cancel out common terms
Now we can cancel \(5^{2n - 2}\) from both the numerator and denominator:
\[ = \frac{5^{3} \cdot 26}{520} \]
This simplifies to:
\[ = \frac{130}{520} = \frac{1}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{25}{4} \text{ or } 6 \frac{1}{4} \]