What is the common difference of an AP in which `a_(18)-a_(14)=32`?

Options

A. 8
B. -8
C. -4
D. 4

Answer

To find the common difference \( d \) of the arithmetic progression (AP) where \( a_{18} - a_{14} = 32 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an AP**:
The nth term of an AP can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

2. **Write the expressions for \( a_{18} \) and \( a_{14} \)**:
Using the formula for the nth term:
\[ a_{18} = a + (18 - 1) \cdot d = a + 17d \]
\[ a_{14} = a + (14 - 1) \cdot d = a + 13d \]

3. **Set up the equation based on the given information**:
According to the problem, we have:
\[ a_{18} - a_{14} = 32 \]
Substituting the expressions for \( a_{18} \) and \( a_{14} \):
\[ (a + 17d) - (a + 13d) = 32 \]

4. **Simplify the equation**:
The \( a \) terms cancel out:
\[ 17d - 13d = 32 \]
This simplifies to:
\[ 4d = 32 \]

5. **Solve for \( d \)**:
Dividing both sides by 4 gives:
\[ d = \frac{32}{4} = 8 \]

### Final Answer:
The common difference \( d \) is \( 8 \).

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