precal question

Question

Use mathematical induction to determine which of the following is divisible by 2 for all natural numbers n. (Select all that apply.)

n² + 9

n(9 + n)

n(n + 2)

n(n + 1)

n² − 2n

Answer 1

#1) for n=2, n^2+9 = 13, so NO

#2)

given Induction Hypothesis says:

n(9+n) = 9n+n^2 is divisible by 2...

then 9(n+1) + (n+1)^2 =

9n + 9 + n^2+2n+1 =

n^2 + 11n + 10 =

n^2+9n + 2n+10 =

n^2+9n + 2(n+10)

By given induction hypothesis, n^2+9n is

divisible by 2. Clearly 2(n+10) is divisible by 2.

[end of proof]

#3) fails on odd integers...

for example, n=1 --> 1(1+2) =3

n=3---> 3(3+2) = 15

#4)

given induction hypothesis says:

n(n+1) = n^2+n is divisible by 2...

then (n+1)(n+2) = n^2+3n+2 = n^2+n + 2n+2

= n^2+n + 2(n+1)

As before, n^2+n is divisible by 2 by induction hypothesis.

Clearly, the second term 2(n+1) is divisible by 2.

[end of proof]

#5) Fails miserably: n=1 ---> 1^2-2(1) = 1-2 = -1 is not even a natural number

n=2 ---> 2^2-2(2) = 0 is arguably a natural number

n=3 ---> 3^2-2(3) = 9-6=3

n=5 --> 5^2-2(5) = 25-10 = 15

in fact, for odd integer n=2k+1,

n^2-2n = (2k+1)^2 - 2(2k+1) = (2k+1)(2k+1 - 2)

= (2k+1)(2k-1)

= 4k^2- 1 is ODD