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

Let first term $=a$ and

Last term $=b$

Middle term of an A.P $=$ A.M of  $a\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}b=\frac{a+b}{2}$

Middle term of $\text{G.P}=\text{G.M\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}a\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}b=\sqrt{ab}$

Middle term of $\text{H.P}=\text{H.M\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}a\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}b=\frac{2ab}{a+b}$

$\left(\frac{a+b}{2}\right)\left(\frac{2ab}{a+b}\right)=ab={\left(\sqrt{ab}\right)}^2$

A,G,H are in G.P

${\text{G}}^2=\text{AH}$

$\therefore$ The form a G.P