Show that x+y is a factor of x^{2n-1}+y^{2n-1} for all natural numbers n.

postillan4

postillan4

Answered question

2021-05-19

Show that x+y is a factor of x2n1+y2n1 for all natural numbers n.

Answer & Explanation

Alara Mccarthy

Alara Mccarthy

Skilled2021-05-20Added 85 answers

Step 1
Let the given statement be P(n)
P(n)=x2n1+y2n1
We check P(n) for n=1
P(1)=x2n1+y2n1=x+y
Thus P(1) is divisible by x+y
Let us assume that P(n) is divisible by x+y when n=kNow for n=k+1 we have
P(k+1)=x2k+1+y2k+1
=(x2)(x2k1)+(y2)(y2k1)
=(x2)(x2k1y2)(x2k1+y2)(x2k1)+(y2)(y2k1)
=(x2y2)(x2k1)+(y2)(y2k1)
=(xy)(x+y)(x2k1)+(y2)(y2k1+x2k1)
Step 2
Now,
(xy)(x+y)(x2k1) and (y2)(y2k1+x2k1) are divisible by x+y
So x2k+1+y2k+1 is also divisible by x+y
So x2n1+y2n1 is divisble by x+y when
n=k+1 if it is divisible by x+y when n=k
So by principle of mathematical induction,
x2n1+y2n1 is divisible by x+y.

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