Let `a_(1), a_(2)`...be positive real numbers in geometric progression. For n, if `A_(n), G_(n), H_(n)` are respectively the arithmetic mean, geometric mean and harmonic mean of `a_(1), a_(2),..., a_(n)`. Then, find an expression for the geometric mean of `G_(1), G_(2),...,G_(n)` in terms of `A_(1), A_(2),...,A_(n), H_(1), H_(2),..., H_(n)`

Explanation

Let `G_(m)` be the geometric mean of `G_(1), G_(2),..,G_(n)`
`rArr 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 GP `a_(1), a_(2),...,a_(n)`
`= [(a_(1).a_(1)...n " times") (r^(1//2).r^(3//3).r^(6//4)...r^(((n-1)n)/(2n)))]^(1//n)`
`= [a_(1)^(n).r^((1)/(2) + (3)/(2) + ...+ (n-1)/(2))]^(1//n)`
`= a_(1) [r^((1)/(2) [((n-1)n)/(2)])]^(1//n) = a_(1) [r^((n-1)/(4))]`...(i)
Now, Now, `A_(n) = (a_(1) + a_(2) + ....+ a_(n))/(n) = (a_(1) (1-r^(n)))/(n(1-r))`
and `H_(n) = (n)/(((1)/(a_(2)) + (1)/(a_(2)) + ...+ (1)/(a_(n))))`
`= (n)/((1)/(a_(1)) (1 + (1)/(r) + ...+ (1)/(r^(n-1))))`
`= (a_(1) n (1-r) r^(n-1))/(1 - r^(n))`
`:. A_(n).H_(n) = (a_(1) (1 -r^(n)))/(n(1-r)) xx(a_(1) n(1 -r)r^(n-1))/((1 -r^(n))) = a_(1)^(2) r^(n-1)`
`rArr underset(k=1)overset(n)prodA_(k) H_(k) = underset(k-1)overset(n)prod (a_(1)^(2) r^(n-1))`
`= (a_(1)^(2). a_(1)^(2).a_(1)^(2)...n " times") xx r^(0).r^(1).r^(2)...r^(n-1)`
`= a_(1)^(2n).r^(1 + 2 + ...+ (n -1))`
`= a_(1)^(2n) r^((n(n-1))/(2)) = [a_(1) r^((n-1)/(4))]^(2n)`
`=[G_(m)]^(2n)` [from Eq.(i)]
`G_(m) = [underset(k=1)overset(n)prod A_(k)H_(k)]^(1//2n)`
`rArr G_(m) = (A_(1) A_(2) ....A_(n) H_(1) H_(2) ...H_(n))^(1//2n)`