If `g_1 and g_2` are two geometric means between a and b, prove that `(g_1^2)/(g_2) + (g_2^2)/(g_1) = a +b`

If g_1 , g_2 are two G.M's between two numbers a and b, then (g_1^2)/(g_2)+(g_2^2)/(g_1) is equal to:

If G_1 is the first of n geometric means between a and b, show that : G_1^(n+1) =a^n b .

If a is the A.M. of b and c and the two geometric means are G_(1) and G_(2) , then prove that G_(1)^(3)+G_(2)^(3) = 2abc

If m is the A.M. of two distinct real numbers l and n""(""l ,""n"">""1) and G1, G2 and G3 are three geometric means between l and n, then G1 ^4+2G2 ^4+G3^ 4 equals, (1) 4l^2 mn (2) 4l^m^2 mn (3) 4l m n^2 (4) 4l^2m^2n^2

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 G is the geometric mean between two distinct positive numbers a and b, then show that 1/(G-a)+1/(G-b)=1/G .

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h

If a,b,c are in G.P. and x,y are the arithmetic means of a,b and b,c respectively. Then prove that a/x+c/y=2 and 1/x+1/y=2/b .

If one G.M., G and two A.M's p and q be inserted between two given numbers, prove that G^(2)= (2p-q) (2q-p)

If g, h, k denotes the side of a pedal triangle, then prove that (g)/(a^(2))+ (h)/(b^(2))+ (k)/(c^(2))=(a^(2)+b^(2) +c^(2))/(2 abc)