Find the common difference of an AP whose first term is 4, the last term is 49 and the sum of all its terms is 265.

Answer

To find the common difference of an arithmetic progression (AP) given the first term, last term, and the sum of all terms, we can follow these steps:

### Step 1: Identify the given values
- First term (a) = 4
- Last term (l) = 49
- Sum of all terms (S_n) = 265

### Step 2: Use the formula for the sum of an AP
The formula for the sum of the first n terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the known values into the formula:
\[ 265 = \frac{n}{2} \times (4 + 49) \]

### Step 3: Simplify the equation
Calculate \( a + l \):
\[ 4 + 49 = 53 \]
Now substitute this back into the equation:
\[ 265 = \frac{n}{2} \times 53 \]

### Step 4: Solve for n
Multiply both sides by 2 to eliminate the fraction:
\[ 530 = n \times 53 \]
Now, divide both sides by 53 to find n:
\[ n = \frac{530}{53} = 10 \]

### Step 5: Use the nth term formula to find the common difference
The nth term of an AP can be expressed as:
\[ T_n = a + (n - 1) \times d \]
Here, we know that the last term (which is the 10th term) is 49:
\[ T_{10} = a + (10 - 1) \times d \]
Substituting the known values:
\[ 49 = 4 + 9d \]

### Step 6: Solve for d
Subtract 4 from both sides:
\[ 49 - 4 = 9d \]
\[ 45 = 9d \]
Now, divide both sides by 9:
\[ d = \frac{45}{9} = 5 \]

### Conclusion
The common difference (d) of the arithmetic progression is **5**.

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