If `n` arithmetic means `A_(1),A_(2),---,A_(n)` are inserted between `a` and `b` then `A_(2)=`

If one geometric mean G and two arithmetic means A_(1)andA_(2) are inserted between two given quantities, then (2A_(1)-A_(2))(2A_(2)-A_(1))=

If one geometric mean G and two arithmetic means A_(1) and A_(2) be inserted between two given quantities,prove that G^(2)=(2A_(1)-A_(2))(2A_(2)-A_(1))

" If A_(1),A_(2),A_(3),...,A_(6) are 'n' numbers inserted between "a,b" to form AP.Then A_(1)+A_(2)+...+A_(n) =

If A_(1), A_(2), A_(3),....A_(51) are arithmetic means inserted between the number a and b, then find the value of ((b + A_(51))/(b - A_(51))) - ((A_(1) + a)/(A_(1) - a))

If a_(1),a_(2),a_(3),......,a_(20) are A.M's inserted between 13 and 67 , then the maximum value of product a_(1)a_(2)a_(3)......a_(20) is

If a_(1), a_(2), a_(3),..., a_(20) are A.M's inserted between 13 and 67 , then the maximum value of product a_(1) a_(2) a_(3)... a_(20) is

If the arithmetic mean of a_(1),a_(2),a_(3),"........"a_(n) is a and b_(1),b_(2),b_(3),"........"b_(n) have the arithmetic mean b and a_(i)+b_(i)=1 for i=1,2,3,"……."n, prove that sum_(i=1)^(n)(a_(i)-a)^(2)+sum_(i=1)^(n)a_(i)b_(i)=nab .

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)

Let A_(1),A_(2),A_(3),"......."A_(m) be arithmetic means between -3 and 828 and G_(1),G_(2),G_(3),"......."G_(n) be geometric means between 1 and 2187. Produmt of geometrimc means is 3^(35) and sum of arithmetic means is 14025. The valjue of n is