The numbers `1,3,6,10,15,21,28"..."` are called triangular numbers. Let `t_(n)` denote the `n^(th)` triangular number such that `t_(n)=t_(n-1)+n,AA n ge 2`.
If `(m+1)` is the `n^(th)` triangular number, then `(n-m)` is

Options

A. (a) `1+sqrt((m^(2)+2m))`
B. (b) `1+sqrt((m^(2)+2))`
C. (c) `1+sqrt((m^(2)+m))`
D. (d) None of these

Explanation

To solve the problem step by step, we will start by recalling the definition of triangular numbers and then derive the required expression.

### Step-by-Step Solution:

1. **Understanding Triangular Numbers**:
The nth triangular number \( t_n \) is defined as:
\[ t_n = t_{n-1} + n \quad \text{for } n \geq 2 \]
The first triangular number \( t_1 = 1 \), and the subsequent numbers are:
\[ t_2 = t_1 + 2 = 1 + 2 = 3 \]
\[ t_3 = t_2 + 3 = 3 + 3 = 6 \]
\[ t_4 = t_3 + 4 = 6 + 4 = 10 \]
Continuing this way, we can express \( t_n \) in terms of the sum of the first \( n \) natural numbers.

2. **General Formula for Triangular Numbers**:
The nth triangular number can be expressed as:
\[ t_n = \frac{n(n + 1)}{2} \]

3. **Given Condition**:
We are given that \( m + 1 = t_n \). Therefore:
\[ m + 1 = \frac{n(n + 1)}{2} \]
Rearranging gives:
\[ m = \frac{n(n + 1)}{2} - 1 \]

4. **Finding \( n - m \)**:
We need to find \( n - m \):
\[ n - m = n - \left(\frac{n(n + 1)}{2} - 1\right) \]
Simplifying this expression:
\[ n - m = n + 1 - \frac{n(n + 1)}{2} \]
To combine the terms, we can express \( n \) as a fraction:
\[ n - m = \frac{2n + 2 - n(n + 1)}{2} \]
Expanding \( n(n + 1) \):
\[ n - m = \frac{2n + 2 - n^2 - n}{2} \]
Simplifying further:
\[ n - m = \frac{-n^2 + n + 2}{2} \]

5. **Final Expression**:
Thus, we have:
\[ n - m = \frac{-n^2 + n + 2}{2} \]

### Conclusion:
The value of \( n - m \) is given by:
\[ n - m = \frac{-n^2 + n + 2}{2} \]