Advertisements
Advertisements
प्रश्न
If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.
उत्तर
\[\text { Let } A_1 , A_2 . . . . . . A_n \text { be n A . M . s between two numbers a and b } . \]
\[\text { Then, } a, A_1 , A_2 . . . . . . . A_n , \text { b are in A . P . with common difference, d } = \frac{b - a}{n + 1} . \]
\[ \therefore A_1 + A_2 + . . . . . . + A_n = \frac{n}{2}\left[ A_1 + A_n \right]\]
\[ = \frac{n}{2}\left[ A_1 - d + A_n + d \right]\]
\[ = \frac{n}{2}\left[ a + b \right]\]
\[ = n \times \left[ \frac{a + b}{2} \right]\]
\[ =\text { A . M . between a and b, which is constant } .\]
APPEARS IN
संबंधित प्रश्न
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))`.
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:
(x − y) and (x + y).
Insert 4 A.M.s between 4 and 19.
Insert 7 A.M.s between 2 and 17.
Insert six A.M.s between 15 and −13.
There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.
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.
If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.
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 the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that:
\[a : b = (2 + \sqrt{3}) : (2 - \sqrt{3}) .\]
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 x, y, z are positive integers then value of expression (x + y)(y + z)(z + x) is ______.
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)`
The minimum value of 4x + 41–x, x ∈ R, is ______.
