Understanding the nth Term and Common Difference in an Arithmetic Progression

In an arithmetic progression (AP), the nth term can be expressed in a general formula:

an a1 (n - 1)d

where a1 represents the first term and d is the common difference between consecutive terms.

Determining the Common Difference

Given that the nth term is expressed as an 4n - 1, we can deduce the common difference by evaluating specific values of n.

Example Calculations

First, let's find the first term when n 1:

a1 4(1) - 1 4 - 1 5

Next, let's determine the second term when n 2:

a2 4(2) - 1 8 - 1 9

Using these values, we can calculate the common difference:

d a2 - a1 9 - 5 4

General Pattern and Verification

To further validate our findings, let's evaluate the third term:

a3 4(3) - 1 12 - 1 13

Indeed, the sequence becomes 5, 9, 13, and so on, with a consistent common difference of 4.

The Common Difference in Terms of n

The common difference in an arithmetic progression can also be understood by observing the coefficient of n in the nth term formula. Here, the coefficient of n is 4, indicating that for each increment in n, the term increases by 4.

Hence, the common difference is 4.

Conclusion

In summary, the common difference in the given arithmetic progression an 4n -1 is 4. This is consistent across all terms and can be determined by evaluating specific values of n, verifying with the general term formula, and recognizing the coefficient of n in the expression.