A series is given as : `1,4,9,16,25,36,…..`
Then the value of `T_(n + 1) - T_(n)` is, where `T_n` is the nth term of the series is:

Options

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

Answer

To solve the problem, we need to find the difference between the (n + 1)th term and the nth term of the given series, where the series is made up of perfect squares: `1, 4, 9, 16, 25, 36, ...`.

### Step-by-Step Solution:

1. **Identify the nth term of the series (T_n)**:
The given series consists of the squares of natural numbers. Therefore, the nth term \( T_n \) can be expressed as:
\[ T_n = n^2 \]

2. **Identify the (n + 1)th term of the series (T_(n + 1))**:
Similarly, the (n + 1)th term \( T_{n + 1} \) is:
\[ T_{n + 1} = (n + 1)^2 \]

3. **Calculate the difference (T_(n + 1) - T_n)**:
Now, we need to find the difference between the (n + 1)th term and the nth term:
\[ T_{n + 1} - T_n = (n + 1)^2 - n^2 \]

4. **Expand the expression**:
Expanding \( (n + 1)^2 \):
\[ (n + 1)^2 = n^2 + 2n + 1 \]
Thus, substituting back into the difference:
\[ T_{n + 1} - T_n = (n^2 + 2n + 1) - n^2 \]

5. **Simplify the expression**:
Now, simplify the expression:
\[ T_{n + 1} - T_n = n^2 + 2n + 1 - n^2 = 2n + 1 \]

### Final Result:
The value of \( T_{n + 1} - T_n \) is:
\[ \boxed{2n + 1} \]