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, and also the ratio of the sum of first $5$ terms to the sum of f

Question

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, and also the ratio of the sum of first $5$ terms to the sum of first $21$ terms.

Accepted Answer

Step 1: $\frac{a + 10d}{a + 17d} = \frac{2}{3}$. Cross-multiplying: $3a + 30d = 2a + 34d \Rightarrow a = 4d$.
Step 2: $\frac{a_5}{a_{21}} = \frac{a + 4d}{a + 20d} = \frac{4d + 4d}{4d + 20d} = \frac{8d}{24d} = \frac{1}{3}$.
Step 3: $\frac{S_5}{S_{21}} = \frac{\frac{5}{2}[2a + 4d]}{\frac{21}{2}[2a + 20d]} = \frac{5[8d + 4d]}{21[8d + 20d]}$.
Step 4: $= \frac{5 	imes 12d}{21 	imes 28d} = \frac{60}{588} = \frac{5}{49}$. So $S_5 : S_{21} = 5 : 49$.

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