Let's call the first sequence number 1.
If you count up the squares of each one you get 8 when n=1, 16 when n=2, 24 when n=3 and so on. The difference between each number is 8 and the expression is 8n.
The second sequence is number 2. If you count up the squares you 8,24,48... When n=1,n=2,n=3... Respectively. The difference between these numbers is 16,24.
The difference between these is 8. So it's a quadratic sequence.
You half the number to get 4.
So the expression starts with 4n^2.
But when n=1 that only equals to 4 so you add 4.
So the expression is 4n^2+4. But that only works for the first one and not for the second or the third. So you have to times the 4 by n.
So the expression is 4n^2 +4n. That works for all the numbers.
The relationship between the sequence 1 and sequence 2 is that when n=1 in sequence 1 the nth termis 8 because the expression is 8n
You times that all by (0.5n+0.5) when n=1. This is because if you divide n=1 in sequence 2 by n=1 in sequence 1 you get 1. 8 divided by 8 is 1.
You do the same for n=2 and the others.
Then you get the sequence 1,1.5,2,2.5 and so on.
The expression for this is 0.5n+0.5.
That is why you times the number by (0.5n+0.5).
So 8*0.5 plus 8*0.5. This gives you 8. Which is the same as n=1 in the sequence 2.
A more simple solution is when n=3 in the sequence 2 it is equal to n=3 + n=2 + n=1 in sequence 1.
To prove it, n=3 in sequence 3 gets you 48 which is equal to 8+16+24.
Written By Ryan Johnson and Eddie Chan.