Question:

The pth term of an A.P is (3p-1) /6. The sum of the first n terms of th A.P is? Explain in detail.

Answer:

To find the sum of the first n terms of an arithmetic progression (A.P), we need to determine the formula for the nth term of the A.P.

Step 1: Finding the formula for the nth term of the A.P:

The pth term of the A.P is given by (3p-1)/6. We can use this information to find the formula for the nth term of the A.P.

Let's substitute p = n into the given expression:
Tn = (3n-1)/6

Step 2: Simplifying the formula for the nth term:

To simplify the formula, we can expand the expression:
Tn = (3n-1)/6
= 3n/6 - 1/6
= n/2 - 1/6

So, the formula for the nth term of the A.P is Tn = n/2 - 1/6.

Step 3: Finding the sum of the first n terms:

To find the sum of the first n terms of the A.P, we can use the formula for the sum of an arithmetic series:

Sn = (n/2)(2a + (n-1)d)

In this case, the first term (a) is given by T1 = (3(1)-1)/6 = 1/2. And the common difference (d) is 1/2.

Substituting these values into the sum formula, we get:
Sn = (n/2)(2(1/2) + (n-1)(1/2))
= (n/2)(1 + n/2 - 1/2)
= (n/2)(n/2)
= n^2 / 4

Therefore, the sum of the first n terms of the A.P is given by Sn = n^2 / 4.

Summary:

- The formula for the nth term of the A.P is Tn = n/2 - 1/6.
- The sum of the first n terms of the A.P is Sn = n^2 / 4.