Proof of Divisibility via Induction: x+a divides x^(2n+1)+a^(2n+1) for all pos. n

[Zibi04]

\(\displaystyle \mbox{Use the principle of mathematical induction to prove that:}\)

. . . . .\(\displaystyle \mbox{a. }\, x^{2n+1}\, +\, a^{2n+1}\, \mbox{ has a factor }\, x\, +\, a\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}\)

. . . . .\(\displaystyle \mbox{b. }\, x^{2n}\, -\, 1\, \mbox{ has a factor }\, x\, +\, 1\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}.\)

Hi all, I've currently been having trouble with the above question (Question 1 a), where I must prove that x^(2n+1) + a^(2n+1) is divisible by x+a for all positive integers n. To be perfectly honest I'm not really sure how to start at all. Any help on the question would be awesome

 

[Deleted member 4993]

\(\displaystyle \mbox{Use the principle of mathematical induction to prove that:}\)

. . . . .\(\displaystyle \mbox{a. }\, x^{2n+1}\, +\, a^{2n+1}\, \mbox{ has a factor }\, x\, +\, a\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}\)

. . . . .\(\displaystyle \mbox{b. }\, x^{2n}\, -\, 1\, \mbox{ has a factor }\, x\, +\, 1\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}.\)

Hi all, I've currently been having trouble with the above question (Question 1 a), where I must prove that x^(2n+1) + a^(2n+1) is divisible by x+a for all positive integers n. To be perfectly honest I'm not really sure how to start at all. Any help on the question would be awesome

Like any "proof by induction":

1) show that the conjecture is true for some value of n₀ (usually n₀= 1)

2) Assume that the conjecture is true for n > n₀

3) Prove that the conjecture is true for (n+1)

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

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[Steven G]

\(\displaystyle \mbox{Use the principle of mathematical induction to prove that:}\)

. . . . .\(\displaystyle \mbox{a. }\, x^{2n+1}\, +\, a^{2n+1}\, \mbox{ has a factor }\, x\, +\, a\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}\)

. . . . .\(\displaystyle \mbox{b. }\, x^{2n}\, -\, 1\, \mbox{ has a factor }\, x\, +\, 1\, \mbox{ for all }\, n\, \in\, \mathbb{Z}^{+}.\)

Hi all, I've currently been having trouble with the above question (Question 1 a), where I must prove that x^(2n+1) + a^(2n+1) is divisible by x+a for all positive integers n. To be perfectly honest I'm not really sure how to start at all. Any help on the question would be awesome

x^(2n+1) + a^(2n+1) is divisible by x+a iff x=-a is a root of x^(2n+1) + a^(2n+1), Do you see that? Personally I feel that is enough but you want to do this by induction. Just use the fact I stated above and the wonderful instructions from Subhotosh and you'll be fine.