Advertisements
Advertisements
Question
If A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then prove that 2A = `(G_1^2)/(G_2) + (G_2^2)/(G_1)`
Solution
Let the two numbers be x and y
∴ A = `(x + y)/2` ......(i)
If G1 and G2 be the geometric means between x and y then x, G1, G2, y are in G.P.
Then y = xr4-1 ....[∵ an = arn–1]
⇒ y = xr3
⇒ `y/x` = r3
⇒ r = `(y/x)^(1/3)`
Now G1 = xr
= `x(y/x)^(1/3)` .....`[because r = (y/x)^(1/3)]`
And G2 = xr2
= `x(y/x)^(2/3)`
∴ From R.H.S. `G_1^2/G_2 + G_2^2/G_1 = (x^2(y/x)^(2/3))/(x(y/x)^(2/3)) + (x^2(y/x)^(4/3))/(x(y/x)^(1/3))`
= `x + x(y/x)^(4/3 - 1/3)`
= `x + x(y/x)`
= x + y = 2A L.H.S. ....[Using equation (i)]
∴ L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are `A+- sqrt((A+G)(A-G))`.
The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour?
What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?
If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the 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))`.
Find the A.M. between:
7 and 13
Find the A.M. between:
12 and −8
Find the A.M. between:
(x − y) and (x + y).
Insert 4 A.M.s between 4 and 19.
Insert six A.M.s between 15 and −13.
Insert A.M.s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.s.
If a is the G.M. of 2 and \[\frac{1}{4}\] , find a.
Find the two numbers whose A.M. is 25 and GM is 20.
Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.
If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.
Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.
If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that \[\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A\]
If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. Then prove that xb – c. yc – a . za – b = 1
If x, y, z are positive integers then value of expression (x + y)(y + z)(z + x) is ______.
