Home
Class 12
MATHS
If a be one A.M and G1 and G2 be then ge...

If a be one A.M and `G_1` and `G_2` be then geometric means between b and c then `G_1^3+G_2^3=`

Text Solution

Verified by Experts

It is given that a is the A.M. of b and c. So,
`a=(b+c)/2` or b+c=2a
Since `G_(1)` and `G_(2)` are two geometric means between b and c, so b,`G_(1),G_(2)`,c is a G.P. with common ratio `r=(c//b)^(1//3)`. Therefore,
`G_(1)=b(c/b)^(1/3),G_(2)=b(c/b)^(2/3)`
`rArrG_(1)^(3)+G_(2)^(3)=b^(2)c+bc^(2)`
=bc(b+c)
=2abc [Using (1)]
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.48|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.49|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.46|1 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If m is the A.M. of two distinct real number l and n(l , n gt 1) and G_1 , G_2 and G_3 are three geometric means between l and n , then G_1^4 + 2G_2^4 + G_3^4 equals.

If m is the A.M of two distinct real number l and n (l,ngt1)andG_(1),G_(2)andG_(3) are three geometric means between l and n , then G_(1)^(4)+2G_(2)^(4)+C_(3)^(4) equals-

If a is the A.M. of b and c and the two geometric mean are G_1 and G_2, then prove that G_1^3+G_2^3=2a b c

If A_1,A_2 be two AM's and G_1,G_2 be the two GM's between two number a and b , then (A_1+A_2)/(G_1G_2) is equal to

If two positive values of a variable be x_(1) and x_(2) and the arithmetic mean be A, geometric mean be G and the harmonic mean be H, then prove that G^(2)=AH

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of m is :

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of n is :

If G_1 ,G_2 are geometric means , and A is the arithmetic mean between two positive no. then show that G_1^2/G_2+G_2^2/G_1=2A .

If a is A.M. of two positive numbers b and c and two G.M's between b and c are G_(1) and G_(2) , prove that G_(1)^(3) + G_(2)^(3) = 2abc .

The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean A and the geometric mean G satisfy the relation 2A + G^2 = 27 . Find the numbers.