Advertisements
Advertisements
प्रश्न
Insert 7 A.M.s between 2 and 17.
उत्तर
Let \[A_1 , A_2 , A_3 , A_4 , A_5 , A_6 , A_7\] be the seven A.M.s between 2 and 17.
Then, 2, \[A_1 , A_2 , A_3 , A_4 , A_5 , A_6 , A_7\] and 17 are in A.P. whose common difference is as follows:
d = \[\frac{17 - 2}{7 + 1}\] = \[\frac{15}{8}\]
\[A_1 = 2 + d = 2 + \frac{15}{8} = \frac{31}{8}\]
\[ A_2 = 2 + 2d = 2 + \frac{15}{4} = \frac{23}{4}\]
\[ A_3 = 2 + 3d = 2 + \frac{45}{8} = \frac{61}{8}\]
\[ A_4 = 2 + 4d = 2 + \frac{15}{2} = \frac{19}{2}\]
\[ A_5 = 2 + 5d = 2 + \frac{75}{8} = \frac{91}{8}\]
\[ A_6 = 2 + 6d = 2 + \frac{45}{4} = \frac{53}{4}\]
\[ A_7 = 2 + 7d = 2 + \frac{105}{8} = \frac{121}{8}\]
Hence, the required A.M.s are
\[\frac{31}{8}, \frac{23}{4}, \frac{61}{8}, \frac{19}{2}, \frac{91}{8}, \frac{53}{4}, \frac{121}{8}\].
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.
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.
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 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.
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 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 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 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 ______.
