Grade 12Algebra
If G1 and G2 are the two G.M.s between b,c and A is their A.M., then show that G1^3 + G2^3=2Abc.
If G1 and G2 are the two G.M.s between b,c and A is their A.M., then show that G1^3 + G2^3=2Abc.
1 Answer
Samyak Jain
G1 and G2 are the two G.M.s between b,c and A is their A.M.
b G1 G2 c . By using basic definition of G.M., we have
G12 = b G2 …...(1) & G22 = c G1 …...........(2)
Multiply (1) by (2)
=>G12 G22 = b G2 c G1 => G1 G2 = bc or G1 = bc / G2 …......(3)
Divide (1) by (2)
=> G12 / G22 = b G2 / c G1 => c G13 = b G23
G13 = b G23 / c …....(4)
From (3), G13 = (bc / G2)3 = b3 c3 / G23 ........(5)
Equate (4) & (5). b G23 / c = b3 c3 / G23
(G23)2 = b2 c4 => G23 = b c2
From (5), G13 = b3 c3 / G23 = b3 c3 / b c2
G13 + G23 = b2 c + b c2 = bc (b + c) ….........(6)
Also, A is the A.M. of b and c. => A = (b + c)/2 => 2A = b + c
Substitute value of (b + c) in (6).
Thus,
G13 + G23 = 2Abc
If satisfied with the solution, pls approve.
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