`S_(n)` be the sum of n terms of the series `(8)/(5)+(16)/(65)+(24)/(325)+"......"`
The value of `S_(8)`, is (a)`(288)/(145)` (b)`(1088)/(545)` (c)`(81)/(41)` (d)`(107)/(245)`

Options

A. `(288)/(145)`
B. `(1088)/(545)`
C. `(81)/(41)`
D. `(107)/(245)`

Explanation

To find the sum of the first 8 terms of the series given by \( S_n = \frac{8}{5} + \frac{16}{65} + \frac{24}{325} + \ldots \), we will follow these steps:

### Step 1: Identify the general term \( T_n \)
The general term of the series can be expressed as:
\[ T_n = \frac{8n}{4n^4 + 1} \]
This can be verified by substituting \( n = 1, 2, 3, \ldots \) to check if it generates the correct terms of the series.

### Step 2: Write the sum of the first \( n \) terms \( S_n \)
The sum of the first \( n \) terms can be derived using the formula:
\[ S_n = 2 \left( 1 - \frac{1}{2n^2 + 2n + 1} \right) \]

### Step 3: Substitute \( n = 8 \) into the formula for \( S_n \)
Now, we will calculate \( S_8 \):
\[ S_8 = 2 \left( 1 - \frac{1}{2(8^2) + 2(8) + 1} \right) \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ 2(8^2) + 2(8) + 1 = 2(64) + 16 + 1 = 128 + 16 + 1 = 145 \]

### Step 5: Substitute back into the equation for \( S_8 \)
Now substituting back:
\[ S_8 = 2 \left( 1 - \frac{1}{145} \right) = 2 \left( \frac{145 - 1}{145} \right) = 2 \left( \frac{144}{145} \right) \]

### Step 6: Simplify the expression
Simplifying the expression gives:
\[ S_8 = \frac{288}{145} \]

### Final Answer
Thus, the value of \( S_8 \) is:
\[ \boxed{\frac{288}{145}} \]

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