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?

Question

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&#039;t get anything. Can you please give me a hint?

tucetiw0 · Accepted Answer

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  )  =  0Then, obviously any one of the following must hold:  a  =  −  b    b  =  −  c    c  =  −  aIn 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.....

Liberty Page · Accepted Answer

HintAt the first step, show that   (  a  +  b  )  (  a  +  c  )  (  b  +  c  )  =  0Next, show that two of the three numbers are opposite.

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?

Answered question

Answer & Explanation

New Questions in Pre-Algebra