Question

Use mathematical induction to prove that the statement is true for every positive integer n.

2 is a factor of n^2 - n + 2..

Answer 1

start by proving its true for n = 1

the assume its true for n = k

Answer 2

if n=1 then
n2−n+2=12−1+2=2

which is a multiple of 2..

Answer 3

now let's assume that n>1 and
2∣(n2−n+2)

we have to prove that
2∣[(n+1)2−(n+1)+2]

Answer 4

(n+1)2−(n+1)+2=n2+2n+1−n−1+2

=(n2−n+2)+2n=2⋅α+2n=2(α+n) and yeah don't forget that....
2∣(n2−n+2)⇔n2−n+2=2⋅α

this the induction hipothesis.

Answer 5

then in assuming its true for n = k you can say that

n^2 - n + 2 = 2p

now prove it for n = k + 1

so you have

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

= k^2 - k + 2 + 2k

using the assumption for 2p

= 2p + 2k

= 2(p + k)

which is divisible by 2

Answer 6

ohk...that's all, thanks for the help

Answer 7

oops should read for n = k

k^2 - k + 2 = 2p

Answer 8

@skylark06 r u frm india

Answer 9

yeah!!

Answer 10

which state in india

Answer 11

@skylark06 u dereee