# Thread: 2, 5, 8, 11, 14... (q2)

1. ## 2, 5, 8, 11, 14... (q2)

Sequence given: 2, 5, 8, 11, 14...

Prove that the product of any two consecutive numbers in the sequence can't be in the sequence?

2. Originally Posted by Natasha
Sequence given: 2, 5, 8, 11, 14...

Prove that the product of any two consecutive numbers in the sequence can't be in the sequence?
The sequence is arithmatic, with a separation of 3 and initial value of 2. So we may model the series as $a_{n}=3n+2$.

We want to multiply two consecutive numbers in the series, so use the n and (n+1) terms:
$a_{n+1} * a_n =(3(n+1)+2)(3n+2)=(3n+5)(3n+2)$
$=9n^2+21n+10$
If this is a member of the series, then $(3q+2)=9n^2+21n+10$ for some integer q. Solving for q we obtain:
$q=3n^2+7n+8/3$ which is never an integer for any n.

-Dan

3. Because each number has form,
$3k+2$
If multiplied together they produce, (of form)
$3k+1$
Impossible by definition of this sequence.
Q.E.D.