The sums of n terms of two arithmetic progresssions are in the ratio `(7n+1):(4n+17).` Find the ratio of their nth terms and also common differences.

Explanation

To solve the problem, we need to find the ratio of the nth terms and the common differences of two arithmetic progressions (APs) given that the sums of their n terms are in the ratio \( (7n + 1):(4n + 17) \).

### Step-by-Step Solution:

**Step 1: Write the formula for the sum of n terms of an AP.**
The sum of the first n terms \( S_n \) of an arithmetic progression can be expressed as:
\[
S_n = \frac{n}{2} \left(2A + (n - 1)D\right)
...