Need genius to help find pattern in these numbers please?

[swimchick1993]

Homework Statement

2 4 3 9 5 25 18 324

Homework Equations

The Attempt at a Solution

2 squared is 4, 3 squared is 9, 5 squared is 25, 18 squared is 324. BUT... what is the next number in the pattern? we squared 2, then 3, then 5, then 18, then...? At first I thought it was prime numbers, but then instead of 7, we jumped to 18?
Help please? I'm going crazy

 

[JG89]

I find it very hard to find a pattern. Here's what I have, even though it may seem a little forced:

lets group the numbers like this: (2,4) (3,9) (5,25) (18,324)

Let's say we start with (2,4) then to get the next pair of numbers we do 4 -2 = 2, then 2 + 1 = 3 (which will be the first number, then square it to get the next one). Then we have (3,9). We now do 9 - 3 = 6, then 6 - 1 = 5. Then we have (5,25). So we do 25 - 5 = 20, then 20 - 2 = 18. Then we have (18,324). So 324 - 18 = 306. Then 306 + 2 = 308, then square that, so we have (308, 94864).

So, we start with (2,4). Let a be the first member of the pair and b be the second member. To get the first member of the second pair we do (b - a) + 1. Then to get the first member of the next pair, (b -a) - 1. The for the next pair, (b - a) - 2. Then for the next pair (b -a) + 2. Then for the next pair (b -a) + 3. Then (b -a) - 3. Then (b -a) - 4 and so on

Again, this may seem a bit forced, but from what you have given me, it's what I can make out of it.

 

[Phrak]

Taking-off on JG's idea (you're a genius, JG) I've introduced a third, and hidden value, m.

By the way, is this legal or mathematically moral?

For a set (n, n^2, m), the following set is (n', n'^2, m').

where, n' = n^2 - n + 1 - m and m' = n'-n+1.

This works, but there should be a way to do it where m isn't hidden.

(2,4,0)
(3,9,2)
(5,25,3)
(18,324,14)

this predicts 293 as the next number.

 

[sutupidmath]

2 4 3 9 5 25 18 324

Or or, which seems quite unlikely but let's give it a try. As i can see the main difficulty is when we want to transit from 4 to 3, from 9 to 5 from 25 to 18 that is let

(2,4),(3,9),(5,25),(18,324),(X,Y) that is we need to find a pattern to generate the first element of each pair.

by taking the difference of the last el. of each pair with the first el. of the upcomming one we get the sequence: 4-3=1,9-5=4,25-18=7,
so, 1,4,7,10,13,16..., would be the sequence of the differences of the previous last el of the pari with the first el. of the upcoming pair.

so it looks to me like the next term of the pair is going to be say X, which is 324-X=10, so X=314. so 98596=(314)^2. so the next pair as far as i am concerned is

(314,98596) and pursuing this pattern it is very easy to generate every other term.

This also looks very logical to me, and there is not any reasonable reason why this wouldn't hold, considering the amount of information that is given in the sequence. If there were more terms, then i would see if this holds or not.

 

[swimchick1993]

WOW, I LOVE ALL THREE OF THESE! AND YOU HAVE EXPANDED MY BRAIN AS WELL! lol! THANK YOU SO MUCH FOR TAKING TIME TO HELP A POOR HIGH SCHOOL FRESHMAN!
I'M ACTUALLY THINKING MORE OUTSIDE THE BOX BECAUSE OF YOUR HELP.
WHAT A GREAT SITE!
THANK YOU

 

[Phrak]

I wonder if she'll discover she has three valid solutions out of an infinitude of them-- or if that's the point of the exercise...

 

[sutupidmath]

I wonder if she'll discover she has three valid solutions out of an infinitude of them-- or if that's the point of the exercise...

Yeah, i am wondering that too, because with the limited amount of information that is provided there, it seems like the three of us managed to find a different pattern for finding next terms, which as far as i am concerned all three of them look logical. However, if we had more information about that sequence, probbably the choices would narrow down.