Find the common difference of an A.P. in which `a_(10)-a_(8)=12`.

Options

A. 0
B. 1
C. 2
D. 6

Answer

To find the common difference of an Arithmetic Progression (A.P.) where \( a_{10} - a_{8} = 12 \), we can follow these steps:

### Step 1: Write the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ a_n = a_1 + (n - 1) \cdot d \]
where \( a_1 \) is the first term and \( d \) is the common difference.

### Step 2: Write the expressions for \( a_{10} \) and \( a_{8} \).
Using the formula for the nth term, we can write:
\[ a_{10} = a_1 + (10 - 1) \cdot d = a_1 + 9d \]
\[ a_{8} = a_1 + (8 - 1) \cdot d = a_1 + 7d \]

### Step 3: Set up the equation based on the given information.
According to the problem, we have:
\[ a_{10} - a_{8} = 12 \]
Substituting the expressions we found:
\[ (a_1 + 9d) - (a_1 + 7d) = 12 \]

### Step 4: Simplify the equation.
Now simplify the left side:
\[ a_1 + 9d - a_1 - 7d = 12 \]
This simplifies to:
\[ 2d = 12 \]

### Step 5: Solve for the common difference \( d \).
To find \( d \), divide both sides by 2:
\[ d = \frac{12}{2} = 6 \]

### Final Answer:
The common difference \( d \) of the A.P. is \( 6 \).

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