Given that a, b, c, . . . and z are positive integers in an arithmetic progression and has a common difference of 4. Which of the following represents the mean of the given integers in term of z?

To find the mean of the given integers in terms of z, we need to determine the value of z in the arithmetic progression.

Since the common difference is 4, we can express the terms of the arithmetic progression as follows:

a = z b = z + 4 c = z + 8 ... z = z + (n-1) * 4

To find the mean, we need to calculate the sum of all the terms and divide it by the total number of terms.

The sum of an arithmetic progression can be calculated using the formula:

Sum = (n/2) * (first term + last term)

In this case, the first term is z and the last term is z + (n-1) * 4. So the sum can be written as:

Sum = (n/2) * (z + z + (n-1) * 4)

Simplifying this expression, we get:

Sum = (n/2) * (2z + (n-1) * 4)

Now, to find the mean, we divide the sum by the total number of terms, which is n:

Mean = Sum / n

Substituting the expression for the sum, we get:

Mean = [(n/2) * (2z + (n-1) * 4)] / n

Simplifying further, we have:

Mean = (2z + (n-1) * 4) / 2

Mean = z + 2(n-1)

Therefore, the mean of the given integers in terms of z is z + 2(n-1).