If a + b = c + d , and a 3 </msup> + b 3 </msup> =

Antoine Hill

Antoine Hill

Answered question

2022-05-26

If a + b = c + d, and a 3 + b 3 = c 3 + d 3 , then prove that:
a 2011 + b 2011 = c 2011 + d 2011

Answer & Explanation

floygdarvn

floygdarvn

Beginner2022-05-27Added 12 answers

Alternatively:
Let x n = a 2 n + 1 + b 2 n + 1 = a ( a 2 ) n + b ( b 2 ) n . Then x n + 2 = ( a 2 + b 2 ) x n + 1 ( a 2 b 2 ) x n and so x n is determined by x 0 and x 1
Let y n = c 2 n + 1 + d 2 n + 1 . Then y n + 2 = ( c 2 + d 2 ) y n + 1 ( c 2 d 2 ) y n and so y n is determined by y 0 = x 0 and y 1 = x 1
We only have to prove that a 2 + b 2 = c 2 + d 2 and a 2 b 2 = c 2 d 2 , which you have done.
Emery Boone

Emery Boone

Beginner2022-05-28Added 3 answers

Show that ( a b ) 2 = ( c d ) 2 . Then consider two cases.
Case 1:
a b = c d .
Case 2:
a b = ( c d ) .
We're given a + b = c + d. Then a = d and b = c. Again, we're done.

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