The sum of n terms of the series
`3/(1^2 * 2^2) + (5)/(2^2 * 3^2) + (7)/(3^2 * 4^2)` + … is :

Options

A. `(n^2 - 2n)/((n-1)^2)`
B. `(n^2 -2n)/((n+1)^2)`
C. `(2n^2 +1)/(n)`
D. none of these

Answer

To solve the problem of finding the sum of the first n terms of the series

\[ \frac{3}{1^2 \cdot 2^2} + \frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \ldots \]

we will first analyze the pattern of the series and then derive the sum.

### Step 1: Identify the general term of the series

The series can be observed as follows:

- The numerator of the k-th term is \(2k + 1\) (for \(k = 1, 2, 3, \ldots\)).
- The denominator of the k-th term is \(k^2 \cdot (k + 1)^2\).

Thus, the k-th term can be expressed as:

\[ T_k = \frac{2k + 1}{k^2 \cdot (k + 1)^2} \]

### Step 2: Calculate the sum of the first two terms

To find the sum of the first n terms, we can start by calculating the sum of the first two terms:

\[ T_1 = \frac{3}{1^2 \cdot 2^2} = \frac{3}{4} \]
\[ T_2 = \frac{5}{2^2 \cdot 3^2} = \frac{5}{36} \]

Now, we add these two terms:

\[ S_2 = T_1 + T_2 = \frac{3}{4} + \frac{5}{36} \]

To add these fractions, we need a common denominator. The least common multiple of 4 and 36 is 36.

Converting \(\frac{3}{4}\) to have a denominator of 36:

\[ \frac{3}{4} = \frac{3 \cdot 9}{4 \cdot 9} = \frac{27}{36} \]

Now we can add:

\[ S_2 = \frac{27}{36} + \frac{5}{36} = \frac{27 + 5}{36} = \frac{32}{36} = \frac{8}{9} \]

### Step 3: Check the options by substituting \(n = 2\)

Now we have \(S_2 = \frac{8}{9}\). We will check which option matches this value when \(n = 2\).

1. **Option 1**: \(4 - \frac{4}{4 - 1^2}\)
- This simplifies to \(4 - \frac{4}{3} = \frac{12 - 4}{3} = \frac{8}{3}\) (not equal to \(\frac{8}{9}\)).

2. **Option 2**: \(4 - \frac{4}{5^2}\)
- This simplifies to \(4 - \frac{4}{25} = \frac{100 - 4}{25} = \frac{96}{25}\) (not equal to \(\frac{8}{9}\)).

3. **Option 3**: \(2 \times 4 \times 8 + 1\)
- This simplifies to \(2 \times 4 \times 8 + 1 = 64 + 1 = 65\) (not equal to \(\frac{8}{9}\)).

4. **Option 4**: (Assuming it is a valid option)
- This option must be checked and found to match \(\frac{8}{9}\).

### Conclusion

After checking the options, we find that the correct answer is **Option D**.