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Question

If a b2c3, a2b3c4, a3b4c5 are in  A.P. (a, b, c > 0) then the minimum value of a + b + c is __


Solution

Observe that a + b + c is a part of AM of a, b, c i.e.,(a+b+c)3 as AM GM & HM,

We can get minimum value of AM from this relation : AM GM

Using  AM GM a + b + c (abc)13 --------------(1)

We need value of abc

Given  a b2 c3, a2 b3 c4, a3 b4 c5 are in  A.P.

2a2b3c4ab2c3 =  ab2c3ab2c3 +  a3b4c5ab2c3

2abc = 1 +  a2 b2 c2

(abc)2 - 2abc + 1 = 0

(abc1)2 = 0

So, abc = 1 

From (1), a + b + c  3

So, the minimum value is 3.

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