What is the value of `(1)/(5xx8)+(1)/(8xx11)+(1)/(11xx14)+...+(1)/(332xx335)`

Options

A. `(66)/(335)`
B. `(22)/(105)`
C. `(22)/(335)`
D. `(11)/(116)`

Answer

To solve the problem, we need to find the value of the series:

\[ S = \frac{1}{5 \times 8} + \frac{1}{8 \times 11} + \frac{1}{11 \times 14} + \ldots + \frac{1}{332 \times 335} \]

### Step 1: Identify the Pattern
Notice that each term can be expressed in a specific form. The general term of the series can be written as:

\[ \frac{1}{(3n + 2)(3n + 5)} \]

where \( n \) starts from 1.

### Step 2: Rewrite Each Term
We can use partial fraction decomposition to rewrite each term:

\[ \frac{1}{(3n + 2)(3n + 5)} = \frac{A}{3n + 2} + \frac{B}{3n + 5} \]

Multiplying through by the denominator \((3n + 2)(3n + 5)\), we get:

\[ 1 = A(3n + 5) + B(3n + 2) \]

### Step 3: Solve for A and B
Expanding the right side gives:

\[ 1 = (3A + 3B)n + (5A + 2B) \]

To satisfy this equation for all \( n \), we set the coefficients equal:

1. \( 3A + 3B = 0 \) (coefficient of \( n \))
2. \( 5A + 2B = 1 \) (constant term)

From the first equation, we have \( A + B = 0 \) or \( B = -A \). Substituting into the second equation:

\[ 5A + 2(-A) = 1 \implies 5A - 2A = 1 \implies 3A = 1 \implies A = \frac{1}{3} \]

Thus, \( B = -\frac{1}{3} \).

### Step 4: Rewrite the Series
Now we can rewrite the original series:

\[ \frac{1}{(3n + 2)(3n + 5)} = \frac{1/3}{3n + 2} - \frac{1/3}{3n + 5} \]

So the series becomes:

\[ S = \frac{1}{3} \left( \left( \frac{1}{5} - \frac{1}{8} \right) + \left( \frac{1}{8} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{14} \right) + \ldots + \left( \frac{1}{332} - \frac{1}{335} \right) \right) \]

### Step 5: Simplify the Series
Notice that this is a telescoping series. Most terms will cancel out:

\[ S = \frac{1}{3} \left( \frac{1}{5} - \frac{1}{335} \right) \]

### Step 6: Calculate the Result
Now we compute:

\[ \frac{1}{5} - \frac{1}{335} = \frac{335 - 5}{5 \times 335} = \frac{330}{1675} \]

Now, substituting back into \( S \):

\[ S = \frac{1}{3} \cdot \frac{330}{1675} = \frac{110}{1675} \]

### Step 7: Final Simplification
We can simplify \( \frac{110}{1675} \):

\[ \frac{110}{1675} = \frac{22}{335} \]

### Final Answer
Thus, the value of the series is:

\[ \boxed{\frac{22}{335}} \]