None 6.x^n+y^n is divisible by x+y if 2 points nis –. (^is denoted for power of) * how​

Question

None
6.x^n+y^n is divisible by x+y if 2 points
nis ……… (^is denoted for
power of) * how​

in progress 0
1 month 2021-08-17T15:13:45+00:00 1 Answer 0 views 0

1. Step-by-step explanation:

The statement to be proved is:

P(n):x

n

−y

n

is divisble by x+y where n is even.

Step 1: Verify that the statement is true for the smallest value of n, here, n=2

P(2):x

2

−y

2

is divisible by x+y

P(2):(x+y)(x−y) is divisible by x+y, which is true.

Therefore P(2) is true.

Step 2: Assume that the statement is true for k

Let us assume that P(k):x

k

−y

k

is divisible by x+y where k is even.

Step 3: Verify that the statement is true for the next possible integer, here for n=k+2

x

k+2

−y

k+2

=x

k+2

−x

2

y

k

+x

2

y

k

−y

k+2

=x

2

(x

k

−y

k

)+y

k

(x

2

−y

2

)

Since (x

k

−y

k

) and (x

2

−y

2

) are both divisible by (x+y), the complete equality is divisible by x+y

Therefore,

P(k+2):x

k+2

−y

k+2

is divisible by x+y where k+2 is even.

Therefore by principle of mathematical induction, P(n) is true.