is divisible by 8 whenever n is an odd positive integer

1) Basic step: P(1) = 1-1 =0 is divisible by 8.

2) inductive step:

hypothesis: is divisible by 8

conclusion: is divisible by 8?

OR

is divisible by 8? <- to make it odd

help me pls

Printable View

- October 16th 2009, 06:05 PMzpwnchenhelp me with Inductive Proof #2
is divisible by 8 whenever n is an odd positive integer

1) Basic step: P(1) = 1-1 =0 is divisible by 8.

2) inductive step:

hypothesis: is divisible by 8

conclusion: is divisible by 8?

OR

is divisible by 8? <- to make it odd

help me pls - October 17th 2009, 04:20 AMtonio

It is P(k+2), of course: if k is an odd positive integer, the next ODD integer is k + 2, not k + 1.

Without induction: n^2 - 1 = (n-)(n+1). As n is odd, exactly one of n-1 or n+1 is divisible by 4 and the other factor is even ==> the whole thing is divisible by 4*2 = 8.

Tonio