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Let a,b,c,d and e be positive real numbe...

Let a,b,c,d and e be positive real numbers such that `a+b+c+d+e=15` and `ab^2c^3d^4e^5=(120)^3xx50`. Then the value of `a^2+b^2+c^2+d^2+e^2` is ___________.

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Verified by Experts

The correct Answer is:
55
`(a + (b)/(2) + (b)/(2) + (c )/(3) + (c )/(3) + (c )/(3) + (d)/(4) + (d)/(4) + (d)/(4) + (d)/(4) + (e )/(5) + (e )/(5) + (e )/(5) + (e )/(5) + (e )/(5))/(15) = 1`
`:. G.M = 1`
`G.M = ((ab^(2) c^(3) d^(4) e^(5))/(a.2^(2).3^(3).4^(4).5^(5)))^(1//5) = 1`
`:. A.M= G.M`
`implies a = (b)/(2) = (c )/(3) = (d)/(4) = (e )/(5) = lambda`
`lambda = 1`
`:. a = 1, b = 2, c = 3, d = 4, e = 5`
`:. a^(2) + b^(2) + c^(2) + d^(2) + e^(2) = 1^(2) 2^(2) + 3^(2) + 4^(2) + 5^(2) = 55`
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