In an A.P., if `t_(18) - t_(14) =32`, then d=

Options

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

Answer

To solve the problem, we need to find the common difference \( d \) in the arithmetic progression (A.P.) given that \( t_{18} - t_{14} = 32 \).

### Step-by-Step Solution:

1. **Understand the terms of A.P.:**
In an arithmetic progression, the \( n \)-th term \( t_n \) can be expressed as:
\[ t_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Write the expressions for \( t_{18} \) and \( t_{14} \):**
- For \( t_{18} \):
\[ t_{18} = a + (18-1)d = a + 17d \]
- For \( t_{14} \):
\[ t_{14} = a + (14-1)d = a + 13d \]

3. **Set up the equation based on the given information:**
We know that:
\[ t_{18} - t_{14} = 32 \]
Substituting the expressions for \( t_{18} \) and \( t_{14} \):
\[ (a + 17d) - (a + 13d) = 32 \]

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

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

### Final Answer:
The value of \( d \) is \( 8 \).

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