Let n in N,n=2k+1,and 1/(a+b+c)=1/a+1/b+1/c. Show that 1/(a^n+b^n+c^n) = 1/(a^n) + (1)/(b^n) + (1)/(c^n) I have tried, but I don't get anything. Can you please give me a hint?

zaiskaladu 2022-09-27 Answered
Let n N , n = 2 k + 1 , a n d   1 a + b + c = 1 a + 1 b + 1 c
Show that 1 a n + b n + c n = 1 a n + 1 b n + 1 c n
I have tried, but I don't get anything. Can you please give me a hint?
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Answers (2)

tucetiw0
Answered 2022-09-28 Author has 12 answers
From the original equation, we get:
a b c = ( a + b + c ) ( a b + b c + c a )
which is equivalent to
a 2 ( b + c ) + b c ( b + c ) + a b ( b + c ) + c a ( b + c ) = 0
( b + c ) ( a + c ) ( a + b ) = 0
Then, obviously any one of the following must hold:
a = b b = c c = a
In any case we can prove the equation
1 a n + b n + c n = 1 a n + 1 b n + 1 c n
with odd n. Since if we take a = b we get
1 ( b ) n + b n + c n = 1 ( b ) n + 1 b n + 1 c n
which is equivalent to
1 c n = 1 c n
and this is true.....
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Liberty Page
Answered 2022-09-29 Author has 3 answers
Hint
At the first step, show that ( a + b ) ( a + c ) ( b + c ) = 0
Next, show that two of the three numbers are opposite.
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