Question

The ratio of the sum of the first \( n \) terms to the sum of the first \( s \) terms of an arithmetic progression is \( r^2 : s^2 \). What is the ratio of its 8th term to the 23rd term of this same progression?

• A. 1 : 3
• B. 2 : 5
• C. 1 : 9
• D. 3 : 10

Hint:

In problems involving ratios of terms in arithmetic progressions, express the terms using the general formula for the \( n \)-th term, and then simplify the ratio.

Answer 1

The Correct Option is A

In an arithmetic progression, the $n$-th term is given by $T_n = a + (n - 1)d$, where $a$ is the first term and $d$ is the common difference.

The ratio of the $8^{\text{th}}$ term to the $23^{\text{rd}}$ term is:
\[ \frac{T_8}{T_{23}} = \frac{a + 7d}{a + 22d} \] Simplify this ratio to get the final answer.