Answer
The common difference d = 0 and the number of terms n = 4.
Step-by-step explanation
We know that the sum S of the first n terms of an arithmetic progression (A.P) is given by the formula:
![S = (n)/(2) * [2a + (n-1)d]](https://img.qammunity.org/2024/formulas/mathematics/college/dz4qptj1vzbcwblxvdig0i5eci6cz7vyoe.png)
where a is the first term, d is the common difference, and n is the number of terms.
In this problem, we are given that S = 20 and a = 8. We need to find d and n.
Substituting the given values in the formula, we get:
![20 = (n)/(2) * [2(8) + (n-1)d]](https://img.qammunity.org/2024/formulas/mathematics/college/51an1uuiywt7a50weougxaxoxnqct7ose2.png)
Simplifying the equation, we get:


We can rewrite this equation as:

Now, we also know that the first term of the A.P is 8. Using this and the formula for the nth term of an A.P, we get:

Substituting the value of d from earlier, we get:

Simplifying this quadratic equation, we get:

Using the quadratic formula, we get:
n = (15 ± √385) / 10
Since n must be a positive integer, we can take the value of n as:
n = (15 + √385) / 10
n ≈ 3.84
Therefore, the number of terms should be rounded up to the nearest integer, which is 4.
Now that we know n = 4, we can substitute it back into the equation for d:



Therefore, the common difference is 0, which means that all the terms in the A.P are the same. In this case, since there are 4 terms, each term must be:
8 / 4 = 2
Therefore, the common difference d = 0 and the number of terms n = 4.