Let a 1 , a 2 ,........, a n be positive real numbers in geometric progression. For each n, let A n , G n , H n be respectively the arithmetic mean, geometric mean & harmonic mean of a 1 , a 2 ,......, a n . Prove that G = 
, Where G is geometric mean between G 1 , G 2 , ........., G n .
Text Solution
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Let G m be the geometric mean of G 1 , G 2 ,..........,G n ⇒ G m = (G 1 · G 2 .....G n ) 1/n
= [(a 1 )(a 1 a 1 r) 1/2 .(a 1 .a 1 r.a 1 r 2 ) 1/3 ... (a 1 .a 1 r.a 1 r 2 ...a 1 r n – 1 ) 1/n ] 1/n
where r is the common ratio of G.P. a 1 , a 2 , .....,a n . = [(a 1 .a 1 ...n times) 
⇒ a 1 . 
= a 1 
. Now, A n = 
= 
and H n = 
= 
= 
Again A n .H n = 
= 
⇒ 
= 
= 
× (r 0 .r 1 .r 2 ...r n – 1 ) = 
= 
= [G m ] 2n ⇒ G m = 
⇒ G m = (A 1 A 2 ... A n .H 1 H 2 ....H n ) 1/2n
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