Find the sum to n terms of the A.P., whose kth term is 5k+1.

Explanation

To find the sum to n terms of the arithmetic progression (A.P.) whose kth term is given by \( a_k = 5k + 1 \), we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The kth term of the A.P. is given by:
\[ a_k = 5k + 1 \]
To find the first term \( a_1 \), we substitute \( k = 1 \):
\[ a_1 = 5(1) + 1 = 5 + 1 = 6 \]

Next, we find the second term \( a_2 \) by substituting \( k = 2 \):
\[ a_2 = 5(2) + 1 = 10 + 1 = 11 \]

Now, we can find the common difference \( d \):
\[ d = a_2 - a_1 = 11 - 6 = 5 \]

### Step 2: Use the formula for the sum of 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:
\[ S_n = \frac{n}{2} (2a + (n - 1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} (2 \cdot 6 + (n - 1) \cdot 5) \]

### Step 3: Simplify the expression
Calculating \( 2a \):
\[ 2a = 2 \cdot 6 = 12 \]
Now substituting this back into the sum formula:
\[ S_n = \frac{n}{2} (12 + (n - 1) \cdot 5) \]
Expanding \( (n - 1) \cdot 5 \):
\[ (n - 1) \cdot 5 = 5n - 5 \]
Now substituting this into the equation:
\[ S_n = \frac{n}{2} (12 + 5n - 5) = \frac{n}{2} (5n + 7) \]

### Step 4: Final expression for the sum
Thus, the sum of the first n terms of the A.P. is:
\[ S_n = \frac{n(5n + 7)}{2} \]

### Summary
The sum to n terms of the A.P. whose kth term is \( 5k + 1 \) is:
\[ S_n = \frac{n(5n + 7)}{2} \]