Question

An A.P., a G.P. and a H.P. have the same first and last terms and the same odd numbers of terms, the middle terms of the three series are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these.

Answer 1

Answer: (B)

G.P.

Answer 2

Let $a$ and $b$ be the same first and last terms of the three series, each having $\left(2n + 1\right)$ terms (i.e., odd number of terms). Then the middle term of the $A.P$. is $\frac{a+b}{2} \left(=A\right)$. the middle term of the $G.P$. is $\sqrt{ab}$(i.e., $G$). The middle term of the $H. P$. is $\frac{2ab}{a+b}$ (i.e., $H$). Since $A, G, H$ are in $G.P$. $\therefore$ the middle terms of the three series are in $G. P$.