Number Sequence

[JWW]

This sequence appeared on a high school math competition and have not been able to figure it out.
What is the 15th number in this sequence and how do you solve it? 4,0,8,40,100,224

 

[Deleted member 4993]

This sequence appeared on a high school math competition and have not been able to figure it out.
What is the 15th number in this sequence and how do you solve it? 4,0,8,40,100,224

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

 

[HallsofIvy]

This sequence appeared on a high school math competition and have not been able to figure it out.
What is the 15th number in this sequence and how do you solve it? 4,0,8,40,100,224

That is a kind of question that appears many times on math tests and does, indeed test one's ability to see "patterns" which is a very important skill in mathematics.

But, strictly speaking, just knowing 6 terms, or any finite number of terms, in a sequence tells you nothing about the rest of the sequence! You have to assume, because we are not told, that there is some simple pattern here! There are some things you can try. Is there a "common difference" so this might be an arithmetic sequence? 0- 4= -4, 8- 0= 8- no, there is no "common difference". Is there a "common ratio" so this is a geometric sequence? 0/8= 0, 8/40= 1/5, 40/100= 0.4= 2/5- no, there is no "common ratio".

It is true that, given any n+ 1 "data points", there is an nth degree polynomial that gives those points. Here, if we think of them as (0, 40), (1, 0), (2, 8), (3, 40), (4, 100), (5, 224), there is a 5th degree polynomial, P(n)= an^5+ bn^4+ cn^3+ dn^3+ en+ f that gives those values. One way to find that polynomial is to put n= 0, 1, 2, 3, 4, and 5 so we have 6 equations to solve for a, b, c, d, e, and f. For example, taking n= 0 we have f= 40. Setting n= 1 a+ b+ c+ d+ e+ f= 0. Setting n= 2, 32a+ 16b+ 8c+ 4d+ 2e+ f= 8, etc. Whether that would give the sequence the person who set this problem was thinking of, I have no idea.

 

[Deleted member 4993]

Sometime ago I saw this sequence:

All those teachers that want the students to guess the next number in the sequence, following should be necessary item:

1, 2, 4, 8, 16, 31, 57, 99, 163, 256,....

This is the number (maximum) of pieces a pie (or a circle) will be cut into with straight cuts. The n th number of this sequence is:

=

 

[stapel]

This sequence appeared on a high school math competition and have not been able to figure it out.
What is the 15th number in this sequence and how do you solve it? 4,0,8,40,100,224

Just FYI: This isn't any of the "standard" sequences, according to OEIS.