Write the n terms of the series`(3)/(7.11^(2))+(5)/(8.12^(2))+(7)/(9.13^(2))+`…...

Answer

To find the n-th term of the series given by

\[ \frac{3}{7 \cdot 11^2} + \frac{5}{8 \cdot 12^2} + \frac{7}{9 \cdot 13^2} + \ldots \]

we will analyze the pattern in both the numerator and the denominator.

### Step 1: Analyze the Numerator
The numerators of the terms are 3, 5, and 7. We can observe that:
- The first term is \(3\) (which can be expressed as \(2 \cdot 1 + 1\)),
- The second term is \(5\) (which can be expressed as \(2 \cdot 2 + 1\)),
- The third term is \(7\) (which can be expressed as \(2 \cdot 3 + 1\)).

From this pattern, we can generalize the numerator for the n-th term as:
\[ \text{Numerator} = 2n + 1 \]

### Step 2: Analyze the Denominator
Now, let’s look at the denominator. The denominators of the terms are \(7 \cdot 11^2\), \(8 \cdot 12^2\), and \(9 \cdot 13^2\).

1. The first part of the denominator is \(7\), \(8\), \(9\):
- The first term is \(7\) (which can be expressed as \(6 + 1\)),
- The second term is \(8\) (which can be expressed as \(6 + 2\)),
- The third term is \(9\) (which can be expressed as \(6 + 3\)).

Thus, we can generalize this part as:
\[ \text{First part of Denominator} = 6 + n \]

2. The second part of the denominator is \(11^2\), \(12^2\), and \(13^2\):
- The first term is \(11^2\) (which can be expressed as \((10 + 1)^2\)),
- The second term is \(12^2\) (which can be expressed as \((10 + 2)^2\)),
- The third term is \(13^2\) (which can be expressed as \((10 + 3)^2\)).

Thus, we can generalize this part as:
\[ \text{Second part of Denominator} = (n + 10)^2 \]

### Step 3: Combine the Parts
Now, we can combine the numerator and the denominator to write the n-th term of the series:

\[ \text{n-th term} = \frac{2n + 1}{(6 + n)(n + 10)^2} \]

### Final Answer
Thus, the n-th term of the series is:

\[ \frac{2n + 1}{(6 + n)(n + 10)^2} \]