The ratio of the $11^{\text{th}}$ term to the $18^{\text{th}}$ term of an $AP$ is $2:3$. Find the ratio of the $5^{\text{th}}$ term to the $21^{\text{st}}$ term,and also the ratio of the sum of the first five terms to the sum of the first $21$ terms.

(A) Let $a$ and $d$ be the first term and common difference of an $AP$.
Given that,$a_{11} : a_{18} = 2 : 3$.
$\Rightarrow \frac{a + 10d}{a + 17d} = \frac{2}{3}$.
$\Rightarrow 3a + 30d = 2a + 34d$.
$\Rightarrow a = 4d$ ....$(i)$.
Now,$a_5 = a + 4d = 4d + 4d = 8d$.
$a_{21} = a + 20d = 4d + 20d = 24d$.
Therefore,$a_5 : a_{21} = 8d : 24d = 1 : 3$.
Now,the sum of the first five terms,$S_5 = \frac{5}{2}[2a + (5-1)d] = \frac{5}{2}[2(4d) + 4d] = \frac{5}{2}(12d) = 30d$.
And the sum of the first $21$ terms,$S_{21} = \frac{21}{2}[2a + (21-1)d] = \frac{21}{2}[2(4d) + 20d] = \frac{21}{2}(28d) = 294d$.
So,the ratio of the sum of the first five terms to the sum of the first $21$ terms is $S_5 : S_{21} = 30d : 294d = 5 : 49$.