If 3rd term of an A.P. is 5 and 7th term is 13, then its common difference is :

Options

A. 1
B. 2
C. 3
D. `4`

Answer

To find the common difference of the arithmetic progression (A.P.) given that the 3rd term is 5 and the 7th term is 13, we can follow these steps:

### Step 1: Understand the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where:
- \( T_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 2: Write equations for the given terms.
From the problem, we know:
- The 3rd term \( T_3 = 5 \)
- The 7th term \( T_7 = 13 \)

Using the formula for the nth term:
1. For the 3rd term:
\[ T_3 = a + (3 - 1) \cdot d = a + 2d = 5 \] (Equation 1)

2. For the 7th term:
\[ T_7 = a + (7 - 1) \cdot d = a + 6d = 13 \] (Equation 2)

### Step 3: Set up the equations.
Now we have two equations:
1. \( a + 2d = 5 \) (Equation 1)
2. \( a + 6d = 13 \) (Equation 2)

### Step 4: Solve the equations.
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 2d) = 13 - 5 \]
This simplifies to:
\[ 6d - 2d = 8 \]
\[ 4d = 8 \]

### Step 5: Find the common difference \( d \).
Now, divide both sides by 4:
\[ d = \frac{8}{4} = 2 \]

### Conclusion:
The common difference \( d \) of the A.P. is **2**.

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