What will be the value of `a_(8) - a_(4)` for the following A.P. 4,9,14,…………., 254

Answer

To find the value of \( a_8 - a_4 \) for the given arithmetic progression (A.P.) 4, 9, 14, ..., 254, we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is:
\[ a = 4 \]
To find the common difference \( d \), we subtract the first term from the second term:
\[ d = a_2 - a_1 = 9 - 4 = 5 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ a_n = 4 + (n - 1) \cdot 5 \]

### Step 3: Simplify the formula for \( a_n \)
Now, we simplify the expression:
\[ a_n = 4 + 5n - 5 = 5n - 1 \]

### Step 4: Calculate \( a_8 \)
Now, we will find \( a_8 \):
\[ a_8 = 5 \cdot 8 - 1 = 40 - 1 = 39 \]

### Step 5: Calculate \( a_4 \)
Next, we will find \( a_4 \):
\[ a_4 = 5 \cdot 4 - 1 = 20 - 1 = 19 \]

### Step 6: Calculate \( a_8 - a_4 \)
Now, we can find \( a_8 - a_4 \):
\[ a_8 - a_4 = 39 - 19 = 20 \]

### Final Answer
The value of \( a_8 - a_4 \) is:
\[ \boxed{20} \]

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