If 18^th and 11^th term of an A.P. are in the ratio 3 : 2, then its 21^st and 5^th terms are in the ratio

Solution (By Examveda Team)

$$\eqalign{ & {18^{{\text{th}}}}\,{\text{term}}:{11^{{\text{th}}}}\,{\text{term}} = 3:2 \cr & \Rightarrow \frac{{{a_{18}}}}{{{a_{11}}}} = \frac{3}{2} \cr & \Rightarrow \frac{{a + 17d}}{{a + 10d}} = \frac{3}{2} \cr & \Rightarrow 2a + 34d = 3a + 30d \cr & \Rightarrow 34d - 30d = 3a - 2a \cr & \Rightarrow a = 4d \cr & {\text{Now,}} \cr & \frac{{{a_{21}}}}{{{a_5}}} = \frac{{a + 20d}}{{a + 4d}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{4d + 20d}}{{4d + 4d}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{24d}}{{8d}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{3}{1} \cr & \therefore {a_{21}}:{a_5} = 3:1 \cr} $$