IIT-JEE 1988Question 1
If the first and the $(2n-1)$st terms of an A.P., a G.P. and an H.P. are equal and their $n$-th terms are $a$, $b$ and $c$ respectively, then
Sequence and Series · Means and Inequalities (AM, GM, HM)
Solution
Formulas used. (1) The $n$-th term is the middle of the $1$st and $(2n-1)$-th terms: A.M., G.M. or H.M. of them(2) A.M. $\ge$ G.M. $\ge$ H.M., and A.M. $\times$ H.M. $=$ G.M.$^2$Step 1 - The three middle terms. Let the common first term be $p$ and the common $(2n-1)$-th term be $q$ (positive): $$a=\frac{p+q}{2},\qquad b=\sqrt{pq},\qquad c=\frac{2pq}{p+q}.$$ Step 2 - Order. By A.M. $\ge$ G.M. $\ge$ H.M.: $a\ge b\ge c$ ✓ (B). Step 3 - Product. $$ac=\frac{p+q}{2}\cdot\frac{2pq}{p+q}=pq=b^2\ \Longrightarrow\ ac-b^2=0\ \checkmark\ (D).$$ (A) needs $p=q$, and (C) fails in general. Answer: (B), (D)