# Which number comes next?

I took a funtrivia quiz that dealt with number patterns.

I got 15/15, but I had to guess on number nine.

When I read the explanation of the answer --I tried to hide the correct responses, but they are all there!-- I found that my guess had nothing to do with the ‘rule’ the author used. Worse, was this comment:

I think I have it figured out, but I’m not sure.

Little help?

-Starting with 1, just keep adding successively higher odd numbers to the preceding number:

-1 + 1 = 0
0 + 3 = 3
3 +5 = 8
8 + 7 = 15
15 + 9 = 24
24 + 11 = 35

etc

Your link to the explanation just led to their front page, but when I look at the pattern, they are simply adding consecutive odd numbers: 1, 3, 5, 7, 9, etc.

I’m guessing the author is referring to the differences between successive terms: 0-(-1)=1, 3-0=3, 8-3=5, 15-8=7, 24-15=9, and so on. That pattern is probably more obvious to some people than the pattern given in the answer.

You have to take the quiz to be able to see the numbers, so here’s what the answer says:

[spoiler]The rule here is to take 1 less than the squares of consecutive whole numbers. So if ‘n’ is a whole number, we take (n^2 - 1):

-1 = 0 - 1 = 0^2 - 1
0 = 1 - 1 = 1^2 - 1
3 = 4 - 1 = 2^2 - 1
8 = 9 - 1 = 3^2 - 1
15 = 16 - 1 = 4^2 - 1
24 = 25 - 1 = 5^2 - 1
35 = 36 - 1 = 6^2 - 1
48 = 49 - 1 = 7^2 - 1
63 = 64 - 1 = 8^2 - 1
80 = 81 - 1 = 9^2 - 1

And so the next number in the series should be 1 less than the square of 10.

x = 10^2 - 1
=> x = 100 - 1
=> x = 99

Hence, the value of x is 99.

This is what I had in mind while writing this question, but if you look closely, there’s a much more obvious pattern that this sequence follows. I won’t spoil it for you and leave it for you to find; that is, unless you have already! I have to admit I hadn’t noticed it myself until a few people wrote to me about it… it’s truly fascinating how the two “rules” produce the same sequence.
[/spoiler]

I think pulykamell and TommyTutone both got the ‘alternate’ answer. Pretty good for not knowing the intended answer

Needless to say, the web site’s answer and the alternate answer are mathematically equivalent, just not obviously so.

One good place to check on such things is the following database of integer sequences:

http://www.research.att.com/~njas/sequences/

Pretty high up on the list of nifty-but-not-quite-obvious mathematical formulas is the fact that the sum of the first n positive odd numbers is n[sup]2[/sup].

This can be proved by mathematical induction, based on the fact that n[sup]2[/sup] + (2n + 1) = (n + 1)[sup]2[/sup]; or a “proof by picture” that involves arranging n[sup]2[/sup] dots in an n by n square, and fitting L-shaped pieces of an odd number of dots over it to form the next bigger square. (I found a picture here along with proofs by people as young as 9 years old (!)).

I guess starting at -1 arbitrarily just threw me… I shouldn’t make assumptions

Although I do see that the original ‘rule’ didn’t start there arbitrarily!

“Much more obvious” is the understatement of the year. It’s blindingly obvious. My 11-year old daughter worked this one out straight away.

Isn’t it obvious that, given a sequence of numbers, there is some function or equation or algorithm that would produce ANY arbitrary value as the next one? That is, there’s no single correct answer - the most you might claim about a particular answer is that the mechanism producing that answer is simpler or more elegant than mechanisms producing any of the other answers. And even this is difficult to claim, because you have to know that no simpler mechanism could be found.

I was thinking along the same lines as the alternate answer, but I was viewing it as: Add the difference of the two previous terms + 2. Same as add by incremental odd numbers, but I think of things oddly. Regardless, I got 99.

Another blindingly obvious pattern…

Notice each number in the sequence is the product of two numbers that differ by two.

So a[sub]n[/sub] = n * (n - 2)

Put in values 1, 2, 3, … for n and you get:

-1,0,3,8,15,24,35,48,63,80, 99, …

I was thinking the alternate pattern was:
(-1)1=-1
0
2=0
13=3
2
4=8
3*5=15

etc. so you continually add 1 to each multiplier to get the next number.

Darn beaten to it. Lance put it nicer than I did.

Yes. And n * (n – 2) = n[sup]2[/sup] – 2n, which is 1 less than n[sup]2[/sup] – 2n + 1 = (n – 1)[sup]2[/sup]. So again, these numbers are 1 less than perfect squares.