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

so i know i need to prove that s1 is correct as the first step

Answer 2

1^2 - 1 + 2
1-1+2
0+2
2

Answer 3

where do i go from here?

Answer 4

correct. now make an assumption
assume its true for n=k
and then we have to prove it's true for n=k+1 based on the assumption

Answer 5

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

Answer 6

yes first we assumed n =k
so \[\rm k^2-k+2\]
2 is a factor of k^2-k+2
and then assumed n=k+1 so yes that's correct simplify that

Answer 7

k^2-k+2 is already simplified

Answer 8

no i mean simplify this `(k+1)^2 - (k+1) + 2`

Answer 9

k^2+1^2-k+1+2
k^2+1-k+3
k^2-k+4

Answer 10

ohhhh wait no! that's not how we should foil (k+1)^2\[\rm (k+1)^2 \cancel{=} k^2+1^2\]

Answer 11

if u need shortcut
then take `square of first term + multiply both term by 2 + square of last term`

Answer 12

lol, i forgot

Answer 13

k^2+2k+1

Answer 14

or draw a box like this