If the first and the $(2 n-1)^{ ext {th }}$ term of an A.P., a G.P. and a H.P. are equal and their

Question

If the first and the $(2 n-1)^{	ext {th }}$ term of an A.P., a G.P. and a H.P. are equal and their $n^{	ext {th }}$ terms are $a, b$ and $c$ respectively then\

Professor PC Thomas Classes · Answer

(a,b,d) : Let the $1^{	ext {st }}$ and $(2 n-1)$ terms be $p$ and $q$.\ Then middle term of the A.P. $=\frac{p+q}{2}=a$\ Middle term of the G.P. $=\sqrt{p q}=b$\ Middle term of the H.P. $=\frac{2 p q}{p+q}=c$\ Now $a c=p q=b^{2} \Rightarrow a c-b^{2}=0$\ Also, A.M. $\geq$ G.M. $\geq$ H.M.\ $\Rightarrow \quad a \geq b \geq c$\ Here $a=b=c$ is true if all the terms are equal.