David is exploring the sequence of numbers given by the rule 2n^2 - 3n + 1

Question

^{David is exploring the sequence of numbers given by the rule 2n^2 - 3n + 1. Prove that the difference between any two consecutive terms from the sequence 2n^2 - 3n + 1 must be an odd number.}

Answer 1

n = 0 N = 1 where N= 2n^2 -3n + 1

n=1 N= 0

n=2 N= 3

n=3 N= 10

for n= 1, 2 or 3, the differences between consecutive N's is an odd number, 1, 3 or 7

assume it's true for n=k

2k^2 -3k + 1

try to show it's also true for 2(k+1)^2 -3(k+1) + 1 = 2(k^2 +2k + 1) - 3k- 3 + 1 = 2k^2 +4k +2 -3k -2 = 2k^2+k+2

for k=0, N= 2

k=1 N= 5

k=2 N= 12 differences are 3 and 7, both odd numbers

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

k=0 4k+1= 1

k=1 4k+1 =5

k=2 4k+1 = 9

k=3 4k+1 = 13 differences are even, always 4

an odd difference plus 4 will always be an odd difference, since an even plus odd always is odd

QED by mathematical induction