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The ratio of the $11^{\text{th}}$ term to $17^{\text{th}}$ term of an A.P. is $3:4$. Find the ratio of $5^{\text{th}}$ term to $21^{\text{st}}$ term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.
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Given $\frac{a + 10d}{a + 16d} = \frac{3}{4}$
$\Rightarrow 4a + 40d = 3a + 48d$
$\Rightarrow a = 8d$ (i)
therefore $\frac{a_5}{a_{21}} = \frac{a + 4d}{a + 20d} = \frac{3}{7}$ using(i)
$a_5: a_{21} = 3:7$
$\frac{S_5}{S_{21}} = \frac{\frac{5}{2} (2a + 4d)}{\frac{21}{2} (2a + 20d)} = \frac{5 \times 20d}{21 \times 36d} = \frac{25}{189}$
Therefore, $S_5:S_{21}=25:189$
$\Rightarrow 4a + 40d = 3a + 48d$
$\Rightarrow a = 8d$ (i)
therefore $\frac{a_5}{a_{21}} = \frac{a + 4d}{a + 20d} = \frac{3}{7}$ using(i)
$a_5: a_{21} = 3:7$
$\frac{S_5}{S_{21}} = \frac{\frac{5}{2} (2a + 4d)}{\frac{21}{2} (2a + 20d)} = \frac{5 \times 20d}{21 \times 36d} = \frac{25}{189}$
Therefore, $S_5:S_{21}=25:189$