Arithmetic Progression (A.P.)

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. The constant difference is called the common difference.

The given sequence is defined by its nth term as 3n/4. We need to determine whether this sequence forms an arithmetic progression and find its common difference.

Proof:

Let's consider the first few terms of the sequence:

First term (n=1): a₁ = 3(1)/4 = 3/4
Second term (n=2): a₂ = 3(2)/4 = 6/4 = 3/2
Third term (n=3): a₃ = 3(3)/4 = 9/4

To determine if the sequence forms an arithmetic progression, we need to check if the difference between any two consecutive terms is constant.

Difference between consecutive terms:

The difference between the second and first terms (a₂ - a₁) is:
(3/2) - (3/4) = (6/4) - (3/4) = 3/4

The difference between the third and second terms (a₃ - a₂) is:
(9/4) - (3/2) = (9/4) - (6/4) = 3/4

As we can see, the difference between any two consecutive terms of the sequence is constant, i.e., 3/4. Therefore, the sequence formed by the nth term 3n/4 is an arithmetic progression.

Common Difference:

The common difference, denoted by 'd', is the difference between any two consecutive terms of an arithmetic progression. In this case, the common difference is 3/4.

Therefore, the sequence defined by its nth term 3n/4 forms an arithmetic progression with a common difference of 3/4.