The sum to n terms of the series `1+ 5 ((4n +1)/(4n - 3)) + 9 ((4n +1) /( 4n -3)) ^(2) + 13 ((4n +1)/( 4n -3)) ^(3) + ((4n + 1 )/( 4n -3)) ^(2)+ …….` is .

Answer

To find the sum to n terms of the given series, we can follow these steps:

### Step 1: Identify the series
The series is given as:
\[ S = 1 + 5 \left( \frac{4n + 1}{4n - 3} \right) + 9 \left( \frac{4n + 1}{4n - 3} \right)^2 + 13 \left( \frac{4n + 1}{4n - 3} \right)^3 + \ldots \]

### Step 2: Determine the general term
The nth term of the series can be expressed as:
\[ T_n = (1 + 4(n - 1)) \left( \frac{4n + 1}{4n - 3} \right)^{n-1} \]
This simplifies to:
\[ T_n = (4n - 3) \left( \frac{4n + 1}{4n - 3} \right)^{n-1} \]

### Step 3: Multiply the series by a factor
To facilitate the summation, we multiply the entire series \( S \) by \( \frac{4n + 3}{4n - 3} \):
\[ \frac{4n + 3}{4n - 3} S = 1 \cdot \frac{4n + 1}{4n - 3} + 5 \cdot \left( \frac{4n + 1}{4n - 3} \right)^2 + 9 \cdot \left( \frac{4n + 1}{4n - 3} \right)^3 + \ldots \]

### Step 4: Set up the equation
Subtract the original series from this new series:
\[ S \left( 1 - \frac{4n + 1}{4n - 3} \right) = \text{(remaining terms)} \]

### Step 5: Simplify the left-hand side
The left-hand side simplifies to:
\[ S \left( \frac{(4n - 3) - (4n + 1)}{4n - 3} \right) = S \left( \frac{-4}{4n - 3} \right) \]

### Step 6: Simplify the right-hand side
The right-hand side consists of terms that can be rearranged and factored:
\[ 1 + 4 \left( \frac{4n + 1}{4n - 3} \right) + 9 \left( \frac{4n + 1}{4n - 3} \right)^2 + \ldots - \text{(last term)} \]

### Step 7: Recognize a geometric series
The remaining terms form a geometric series where:
- First term \( a = 1 \)
- Common ratio \( r = \frac{4n + 1}{4n - 3} \)

### Step 8: Use the formula for the sum of a geometric series
The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Thus, substituting \( a \) and \( r \):
\[ S = \frac{4}{4n - 3} \left( \frac{(4n + 1)^n - 1}{(4n + 1) - (4n - 3)} \right) \]

### Step 9: Solve for S
Now, we can isolate \( S \):
\[ S = \frac{(4n - 3)((4n + 1)^n - 1)}{4} \]

### Step 10: Final expression
After simplification, we find:
\[ S = n(4n - 3) \]