The sums of n terms of two arithmetic progressions are in the ratio (7n-5) : ( 5n+17) . Show that their 6th terms are equal.

Explanation

To solve the problem, we need to show that the sixth terms of two arithmetic progressions (APs) are equal given that the sums of their n terms are in the ratio \( (7n - 5) : (5n + 17) \).

### Step-by-Step Solution:

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