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The ratio of the $11^{th}$ term to the $18^{th}$ term of an A.P. is $2: 3$. Find the ratio of the $5^{th}$ term to the $21^{st}$ term. Also, find the ratio of the sum of first $5$ terms to the sum of first $21$ terms.
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$\frac{a + 10d}{a + 17d} = \frac{2}{3}$
$3a +30d = 2a + 34d \Rightarrow a = 4d$
Therefore, $\frac{a + 4d}{a + 20d} = \frac{4d + 4d}{4d + 20d} = \frac{8d}{24d} = \frac{1}{3}$
$\frac{S_5}{S_{21}} = \frac{\frac{5}{2}[2a + 4d]}{\frac{21}{2}[2a +20d]} = \frac{5[8d + 4d]}{21[8d + 20d]}$
$= \frac{5 \times 12d}{21 \times 28d} = \frac{5}{49}$ or $S_5: S_{21} = 5:49$
$3a +30d = 2a + 34d \Rightarrow a = 4d$
Therefore, $\frac{a + 4d}{a + 20d} = \frac{4d + 4d}{4d + 20d} = \frac{8d}{24d} = \frac{1}{3}$
$\frac{S_5}{S_{21}} = \frac{\frac{5}{2}[2a + 4d]}{\frac{21}{2}[2a +20d]} = \frac{5[8d + 4d]}{21[8d + 20d]}$
$= \frac{5 \times 12d}{21 \times 28d} = \frac{5}{49}$ or $S_5: S_{21} = 5:49$