If ` G_(1)` is the first of n G.M. s between positive numbers a and b , then show that ` G_(1)^(n+1) = a^(n) b ` .

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

Suppose p is the first of n(n > 1) AM's between two positive numbers a and b, then value of p is (i)(na+b)/(n+1) (ii)(na-b)/(n+1) (iii)(nb+a)/(n+1) (iv) (nb-a)/(n+1)

If the ratio of A.M. and G.M. of two positive numbers a and b is m : n, then prove that : a:b=(m+sqrt(m^(2)-n^(2))):(m-sqrt(m^(2)-n^(2)))

If G_(1) . G_(2) , g_(3) are three geometric means between two positive numbers a and b , then g_(1) g_(3) is equal to

Let 3 geometric means G_(1),G_(2),G_(3) are inserted between two positive number a and b such that (G_(3)-G_(2))/(G_(2)-G_(1))=2 then (b)/(a) equal to (i)2 (ii) 4 (iii) 8 (iv) 16

If g_r is the r_(th) term of a G.P. with g_1=a nad g_2=b , then prove that sum_(r=1)^n g_1=(b.g_n-a^2)/(b-a)

If m be the A.M. and g be the G.M. of two positive numbers 'a' and 'b' such that m:g=1:2 , then (a)/(b) can be equal to

If a is the A.M. between b and c, b the G.M. between a and c, then show that 1/a,1/c,1/b are in A.P.

If both the A.M. between m and n and G.M. between two distinct positive numbers a and b are equal to (ma+nb)/(m+n) , then n is equal to