The ratio of the $11^{th}$ term to the $17^{th}$ term of an A.P. is $3:4$. Find the ratio of the $5^{th}$ term to the $21^{st}$ term of the same A.P. Also, find the ratio of the sum of first 5 terms t

Question

The ratio of the $11^{th}$ term to the $17^{th}$ term of an A.P. is $3:4$. Find the ratio of the $5^{th}$ term to the $21^{st}$ term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

Accepted Answer

Step 1: Given $\frac{a + 10d}{a + 16d} = \frac{3}{4}$
$\Rightarrow 4a + 40d = 3a + 48d$
$\Rightarrow a = 8d$ ... (i)
Step 2: Therefore $\frac{a_5}{a_{21}} = \frac{a + 4d}{a + 20d} = \frac{12d}{28d} = \frac{3}{7}$ using (i)
Step 3: $a_5 : a_{21} = 3 : 7$
Step 4: $\frac{S_5}{S_{21}} = \frac{\frac{5}{2}(2a + 4d)}{\frac{21}{2}(2a + 20d)} = \frac{5 	imes 20d}{21 	imes 36d} = \frac{25}{189}$

Question

Solution