If the numbers `n-3,4n-2,5n+1` are in AP, what is the value of n?
(a)1
(b)2
(c)3
(d)4

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the value of \( n \) such that the numbers \( n-3 \), \( 4n-2 \), and \( 5n+1 \) are in Arithmetic Progression (AP), we can use the property of AP that states the middle term is equal to the average of the other two terms. This can be expressed mathematically as:

\[ \text{Middle term} = \frac{\text{First term} + \text{Third term}}{2} \]

### Step-by-Step Solution:

1. **Identify the terms**:
- First term \( A = n - 3 \)
- Second term \( B = 4n - 2 \)
- Third term \( C = 5n + 1 \)

2. **Set up the equation for AP**:
\[ B = \frac{A + C}{2} \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ 4n - 2 = \frac{(n - 3) + (5n + 1)}{2} \]

3. **Simplify the right-hand side**:
Combine the terms in the numerator:
\[ 4n - 2 = \frac{n - 3 + 5n + 1}{2} \]
\[ 4n - 2 = \frac{6n - 2}{2} \]
Simplifying further:
\[ 4n - 2 = 3n - 1 \]

4. **Rearrange the equation**:
Move \( 3n \) to the left side:
\[ 4n - 3n = -1 + 2 \]
\[ n = 1 \]

5. **Conclusion**:
The value of \( n \) is \( 1 \).

### Final Answer:
The value of \( n \) is \( 1 \) (Option a).

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