Given:
z = 1 + ai is a complex number, where a > 0.
z^3 is a real number.

To Find:
The sum of 1, z, z^2, ..., z^11.

Solution:
To find the sum of 1, z, z^2, ..., z^11, we first need to determine the values of z and z^3.

Finding z:
We are given that z = 1 + ai, where a > 0.
To find the value of z, we can equate the real and imaginary parts of z to 1 and a, respectively.

Real Part:
1 = 1 (since the real part of 1 is 1)

Imaginary Part:
0 = a (since the imaginary part of 1 is 0)

Therefore, z = 1 + 0i = 1.

Finding z^3:
We are given that z^3 is a real number.
Since z = 1, z^3 = 1^3 = 1.

Sum of 1, z, z^2, ..., z^11:
Now that we know the values of z and z^3, we can find the sum of 1, z, z^2, ..., z^11.

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 1 and the last term is z^11.
Since z = 1, z^11 = 1^11 = 1.

Using the formula, we have:
Sum = (11/2)(1 + 1) = 11.

Therefore, the sum of 1, z, z^2, ..., z^11 is equal to 11.

Summary:
- Given z = 1 + ai, where a > 0, and z^3 is a real number.
- Found the value of z by equating the real and imaginary parts to 1 and a, respectively.
- Found z^3 = 1.
- Calculated the sum of 1, z, z^2, ..., z^11 using the formula for the sum of an arithmetic progression.
- The sum is equal to 11.