Number of sub parts into which ‘n’ straight lines in a plane can divide it is: (a) `(n^(2)+n+2)/(2)` (b) `(n^(2)+n+4)/(2)` (c) `(n^(2)+n+6)/(2)` (d) none

Options

A. `(n^(2)+n+2)/(2)`
B. `(n^(2)+n+4)/(2)`
C. `(n^(2)+n+6)/(2)`
D. none

Answer

To find the number of sub-parts into which 'n' straight lines in a plane can divide it, we can follow these steps:

### Step 1: Understand the Problem
When we draw straight lines in a plane, they can intersect each other. Each intersection can create additional regions or parts in the plane. We need to find a formula that gives us the total number of regions created by 'n' lines.

### Step 2: Analyze the Cases
1. **For 1 line (n = 1)**: A single line divides the plane into 2 parts.
- \( L_1 = 2 \)

2. **For 2 lines (n = 2)**: If the two lines intersect, they divide the plane into 4 parts.
- \( L_2 = 4 \)

3. **For 3 lines (n = 3)**: If all three lines intersect each other, they divide the plane into 7 parts.
- \( L_3 = 7 \)

### Step 3: Identify a Pattern
From the above cases, we can see:
- \( L_1 = 2 \)
- \( L_2 = 4 \)
- \( L_3 = 7 \)

Now, let's calculate the differences:
- \( L_2 - L_1 = 4 - 2 = 2 \)
- \( L_3 - L_2 = 7 - 4 = 3 \)

Continuing this pattern, we can see that:
- \( L_4 - L_3 = 4 \) (if we add a fourth line that intersects all previous lines)

### Step 4: Generalize the Formula
From the differences, we can observe that:
- \( L_n - L_{n-1} = n \)

This means that the number of new regions created by adding the nth line is equal to n.

### Step 5: Summation
To find a general formula, we can sum the contributions from each line:
- \( L_n = L_1 + (L_2 - L_1) + (L_3 - L_2) + ... + (L_n - L_{n-1}) \)
- This can be expressed as:
\[ L_n = 2 + 2 + 3 + 4 + ... + n \]

### Step 6: Use the Formula for Sum of First n Natural Numbers
The sum of the first n natural numbers is given by:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Thus, we can write:
\[ L_n = 2 + \left(\frac{(n-1)n}{2}\right) \]

### Step 7: Final Formula
After simplifying, we find:
\[ L_n = \frac{n^2 + n + 2}{2} \]

### Conclusion
Thus, the number of sub-parts into which 'n' straight lines in a plane can divide it is given by:
\[ \boxed{\frac{n^2 + n + 2}{2}} \]