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

Answers ( )

    0
    2021-08-17T15:15:13+00:00

    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.

Leave an answer

Browse
Browse

18:9+8+9*3-7:3-1*13 = ? ( )