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`.
The number of positive integers lying between `t_(100)` and `t_(101)` are:

Options

A. (a) `99`
B. (b) `100`
C. (c) `101`
D. (d) `102`

Explanation

To find the number of positive integers lying between the 100th triangular number \( t_{100} \) and the 101st triangular number \( t_{101} \), we can follow these steps:

### Step 1: Understand the formula for triangular numbers
The \( n^{th} \) triangular number \( t_n \) is defined as:
\[ t_n = t_{n-1} + n \quad \text{for } n \geq 2 \]
Given that \( t_1 = 1 \), we can calculate the first few triangular numbers:
- \( t_2 = t_1 + 2 = 1 + 2 = 3 \)
- \( t_3 = t_2 + 3 = 3 + 3 = 6 \)
- \( t_4 = t_3 + 4 = 6 + 4 = 10 \)
- \( t_5 = t_4 + 5 = 10 + 5 = 15 \)
- Continuing this, we can find \( t_{100} \) and \( t_{101} \).

### Step 2: Calculate \( t_{100} \) and \( t_{101} \)
Using the formula:
\[ t_{101} = t_{100} + 101 \]
This means that the difference between \( t_{101} \) and \( t_{100} \) is 101.

### Step 3: Find the range of integers between \( t_{100} \) and \( t_{101} \)
The integers lying between \( t_{100} \) and \( t_{101} \) are:
\[ t_{100} + 1, t_{100} + 2, \ldots, t_{101} - 1 \]
The total count of integers from \( t_{100} + 1 \) to \( t_{101} - 1 \) can be calculated as:
\[ (t_{101} - 1) - (t_{100} + 1) + 1 = t_{101} - t_{100} - 1 \]
Substituting \( t_{101} - t_{100} = 101 \):
\[ = 101 - 1 = 100 \]

### Conclusion
Thus, the number of positive integers lying between \( t_{100} \) and \( t_{101} \) is \( 100 \).

### Final Answer
The answer is \( 100 \).

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