Find the value of `a_(25)- a_(15)` for the AP: 6, 9, 12, 15, .....

Answer

To find the value of \( a_{25} - a_{15} \) for the arithmetic progression (AP) given as 6, 9, 12, 15, ..., we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the AP is 6, and the common difference \( d \) can be calculated as follows:
\[ d = 9 - 6 = 3 \]

### Step 2: Write the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an AP is given by:
\[ a_n = a + (n - 1) \cdot d \]

### Step 3: Calculate \( a_{25} \)
Using the formula for the 25th term:
\[ a_{25} = a + (25 - 1) \cdot d = 6 + (24) \cdot 3 \]
Calculating this:
\[ a_{25} = 6 + 72 = 78 \]

### Step 4: Calculate \( a_{15} \)
Now, using the formula for the 15th term:
\[ a_{15} = a + (15 - 1) \cdot d = 6 + (14) \cdot 3 \]
Calculating this:
\[ a_{15} = 6 + 42 = 48 \]

### Step 5: Find \( a_{25} - a_{15} \)
Now we can find the difference:
\[ a_{25} - a_{15} = 78 - 48 = 30 \]

### Final Answer
Thus, the value of \( a_{25} - a_{15} \) is \( \boxed{30} \).

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