In a geometric progression, the sum of the first n terms is 3069 and the first term is 3. If the nth term in the progression is 1536, find the common ratio

Sure, here are the steps to solve the problem:

1. In a geometric progression, the nth term (Tn) can be calculated using the formula Tn = a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number.

2. From the problem, we know that Tn = 1536 and a = 3. We can substitute these values into the formula to get 1536 = 3 * r^(n-1).

3. Simplifying this equation gives us r^(n-1) = 1536 / 3 = 512.

4. The sum of the first n terms in a geometric progression can be calculated using the formula S_n = a * (r^n - 1) / (r - 1).

5. From the problem, we know that S_n = 3069. We can substitute this value, along with the values for a and r^(n-1), into the formula to get 3069 = 3 * (512*r - 1) / (r - 1).

6. Solving this equation for r will give us the common ratio of the geometric progression.

Please note that this is a complex equation to solve, and you may need to use numerical methods or a software tool to find the exact value of r.