Problem: The pth term of an AP is 3p-1/6. Find the sum of first n terms of the AP.

Solution:

Step 1: Find the common difference of the AP.

The nth term of an AP is given by the formula tn = a + (n-1)d, where a is the first term, d is the common difference, and n is the term number.

Substituting the given expression for tn, we get:

3p-1/6 = a + (p-1)d

Multiplying both sides by 6, we get:

18p - 1 = 6a + 6(p-1)d

Simplifying, we get:

6d = 18

d = 3

Therefore, the common difference of the AP is 3.

Step 2: Find the first term of the AP.

Substituting p = 1 in the expression for tn, we get:

t1 = 3(1) - 1/6 = 2.5

Therefore, the first term of the AP is 2.5.

Step 3: Find the sum of first n terms of the AP.

The sum of first n terms of an AP is given by the formula Sn = n/2(2a + (n-1)d), where a is the first term, d is the common difference, and n is the number of terms.

Substituting the values we have found, we get:

Sn = n/2(2(2.5) + (n-1)3)

Simplifying, we get:

Sn = 1.5n^2 + 0.5n

Therefore, the sum of first n terms of the AP is 1.5n^2 + 0.5n.

Answer: The sum of first n terms of the given AP is 1.5n^2 + 0.5n.