find whether x^n + y^n is divisible by x-y and y= 0 or not
Answers
Step-by-step explanation:
Let P(n) denote the statement
x^n-y^nx
n
−y
n
is divisible by x-y
put n=1,
P(1): x - y is divisible by x-y
Assume the result is true for n=k
That is, x^k-y^kx
k
−y
k
is divisible by x-y
\frac{x^k-y^k }{x-y}= c
x−y
x
k
−y
k
=c
where c is an integer
\begin{lgathered}x^k-y^k=c(x-y)\\\\x^k=y^k+c(x-y)\end{lgathered}
x
k
−y
k
=c(x−y)
x
k
=y
k
+c(x−y)
To prove P(k+1) is true
That is to prove:
X^{k+1}-y^{k+1}X
k+1
−y
k+1
is divisible by x-y
\begin{lgathered}X^{k+1}-y^{k+1}\\\\=x^k.x-y^{k+1}\\\\=[y^k+c(x-y)]x-y^{k+1}\\\\=xy^k+cx(x-y)-y^k.y\\\\=(x-y)y^k+cx(x-y)\\\\=(x-y)(y^k+cx)\end{lgathered}
X
k+1
−y
k+1
=x
k
.x−y
k+1
=[y
k
+c(x−y)]x−y
k+1
=xy
k
+cx(x−y)−y
k
.y
=(x−y)y
k
+cx(x−y)
=(x−y)(y
k
+cx)
, is divisible by x-y
Therefore P(k+1) is true
Hence by mathematical induction P(n) is true for all natural numbers.