Find the sum of 28 terms of an A.P. whose nth term is 8n -5.

Answer

To find the sum of the first 28 terms of the arithmetic progression (A.P.) whose nth term is given by the formula \( T_n = 8n - 5 \), we can follow these steps:

### Step 1: Determine the first term (a)
The first term \( a \) can be found by substituting \( n = 1 \) into the nth term formula:
\[ a = T_1 = 8(1) - 5 = 8 - 5 = 3 \]

### Step 2: Determine the second term (T2) to find the common difference (d)
Next, we find the second term by substituting \( n = 2 \):
\[ T_2 = 8(2) - 5 = 16 - 5 = 11 \]
The common difference \( d \) is calculated as:
\[ d = T_2 - T_1 = 11 - 3 = 8 \]

### Step 3: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first \( n \) terms \( S_n \) of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Here, \( n = 28 \), \( a = 3 \), and \( d = 8 \).

### Step 4: Substitute the values into the formula
Substituting the values into the sum formula:
\[ S_{28} = \frac{28}{2} \times (2 \times 3 + (28 - 1) \times 8) \]
Calculating further:
\[ S_{28} = 14 \times (6 + 27 \times 8) \]
Calculating \( 27 \times 8 \):
\[ 27 \times 8 = 216 \]
So,
\[ S_{28} = 14 \times (6 + 216) = 14 \times 222 \]

### Step 5: Calculate \( 14 \times 222 \)
Now, we compute:
\[ S_{28} = 14 \times 222 \]
Calculating \( 14 \times 222 \):
\[ 14 \times 222 = 3108 \]

### Final Answer
Thus, the sum of the first 28 terms of the A.P. is:
\[ \boxed{3108} \]