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If a + b + c = 0, then (a5 + b5 + c5)/(a2 + b2 + c2)(a3 + b3 + c3) = m/n, find the value of m + n

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2 Answers

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by (475 points)

We are using some formula of algebraThen solve it by putting some attention on the question

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edited by

Correct option is B.

a + b + c = 0

\(\Rightarrow\) a + b = - c or b + c = -a or a + c = - b ...... (1)

If a + b + c = 0

Then (a + b + c)2 = 0

\(\Rightarrow\) a2 + b2 + c2 + 2(ab + bc + ac) = 0

\(\Rightarrow\) a2 + b​​​​​​​2 + c​​​​​​​2 = - 2(ab + bc + ac) ........ (2)

Also If a + b + c = 0

Then a3 + b​​​​​​​3 + c​​​​​​​3 = 3abc .......... (3)

Now, (a3 + b​​​​​​​3 + c​​​​​​​3) (a2 + b​​​​​​​2 + c​​​​​​​2)

= (a5 + b​​​​​​​5 + c​​​​​​​5) + (a3b2 + a3c2 + b3a2 + b3c2 + c3a2 + c3b2)

= (a5 + b​​​​​​​5 + c​​​​​​​5) + (a3b2 + b3a2 + b3c2 + c3b2 + a3c2 + c3a2)

= (a5 + b​​​​​​​5 + c​​​​​​​5) + (a2b​​​​​​​2(a + b) + b2c​​​​​​​2(b + c) + a2c​​​​​​​2(a + c))

= (a5 + b​​​​​​​5 + c​​​​​​​5) + (- a2b​​​​​​​2c - b2c​​​​​​​2a - a2c​​​​​​​2b) (from (1))

= (a5 + b​​​​​​​5 + c​​​​​​​5) - abc (ab + bc + ac)

\(\therefore\) a5 + b​​​​​​​5 + c​​​​​​​5 = (a3 + b​​​​​​​3 + c​​​​​​​3) (a2 + b​​​​​​​2 + c​​​​​​​2) + abc (ab + bc + ac)

= (a3 + b​​​​​​​3 + c​​​​​​​3) (a2 + b​​​​​​​2 + c​​​​​​​2) + \(\frac{(a^3 + b^3 + c^3)}{3} \times \frac{(a^2 + b^2 + c^2)}{-2}\)

(From (2) & (3))

\(= \frac{5}{6}\)(a3 + b​​​​​​​3 + c​​​​​​​3) (a2 + b​​​​​​​2 + c​​​​​​​2)

\(\Rightarrow\) \(\frac{a^5 + b^5 + c^5}{(a^2 + b^2 + c^2)(a^3 + b^3 + c^3)}\) \(= \frac{5}{6}\) \(= \frac{m}{n}\) (Given)

\(\therefore\) m = 5 & n = 6

\(\Rightarrow\) m + n = 5 + 6 = 11.

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