A.M. between two distinct positive numbers is n(>1) times the G.M. between them. Show that the numbers are in the ratio `(n+sqrt(n^2-1)):(n-sqrt(n^2-1))`.

The A.M. between two distinct positive numbers is twice the G.M. between them. Find the ratio of the greater to the smaller.

The sum of two numbers is 6 times their G.M. show that numbers are in the ratio 3+2sqrt2: 3-2sqrt2 .

The A.M. between two positive numbers is 34 and their G.M. is 16. find the numbers.

If the geometric mea is (1)/(n) times the harmonic mean between two numbers, then show that the ratio of the two numbers is 1+sqrt(1-n^(2)):1-sqrt(1-n^(2)) .

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

Prove that the sum of n arithmetic means between two numbers is n times the single A.M. between them.

Show that the sum of first n even numbers is equal to (1+1/n) times the sum of first n odd numbers.

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

The sequence N of natural numbers is divided into classes as follows : Show that the sum of the numbers in the nth row is n(2n^2+1) .

If (a^n+b^n)/(a^(n-1)+b^(n-1)) is the A.M. between a and b, then find the value of n.