Step 1: Understanding the Concept:
Given the ratio of sums \( S_n / S'_n \), the ratio of the \( k\text{-th} \) terms \( a_k / a'_k \) can be found by substituting \( n = 2k - 1 \). Step 2: Detailed Explanation:
Let the ratio of sums be \( \frac{S_n}{S'_n} = \frac{2n + 3}{3n - 1} \).
We know that \( \frac{S_n}{S'_n} = \frac{\frac{n}{2}[2a + (n-1)d]}{\frac{n}{2}[2A + (n-1)D]} = \frac{a + (\frac{n-1}{2})d}{A + (\frac{n-1}{2})D} \).
We want the ratio of the 5th terms: \( \frac{a_5}{A_5} = \frac{a + 4d}{A + 4D} \).
Comparing the expressions, we need \( \frac{n-1}{2} = 4 \).
\[ n - 1 = 8 \implies n = 9 \]
Substitute \( n = 9 \) into the given ratio:
\[ \text{Ratio} = \frac{2(9) + 3}{3(9) - 1} \]
\[ \text{Ratio} = \frac{18 + 3}{27 - 1} = \frac{21}{26} \]. Step 3: Final Answer:
The ratio of the 5th terms is 21:26.