Question 155130: I need to know what number comes next in this sequence and why.
9,73,241,561,1081,1849,_?__
Found 3 solutions by stanbon, Edwin McCravy, terryriegel:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! I need to know what number comes next in this sequence and why.
9,73,241,561,1081,1849,_?__
-----------------------------
I matched 1,2,3,...,6 up with the numbers.
I tried a linear regression, a quadratic regression,
and a cubic regression on it. Best results were
with the cubic:
y = 8x^3 + 4x^2 -4x = 1
-----------------------------
The next term would be f(7) = 2913
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Cheers,
Stan H.
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website! Edwin's solution
by a difference table:
Stanbon's solution is correct but it requires
something you probably have not studied:
We make what is called a "difference table":
Write the numbers in a row:
9 73 241 561 1081 1849
Under that row, subtract each adjacent pair of numbers,
placing the difference between and below them.
(73-9=64, so write 64 between and below the pair 9 73,
241-73=168, so write 168 between and below the pair 73 241, etc.)
Like this:
9 73 241 561 1081 1849
64 168 320 520 768
That row is called the row of 1st differences.
Under that row, again subtract each adjacent pair of numbers,
placing the difference between and below them, like this:
9 73 241 561 1081 1849
64 168 320 520 768
104 152 200 248
That row is called the row of 2nd differences.
Under that row, again subtract each adjacent pair of numbers,
placing the difference between and below them, like this:
9 73 241 561 1081 1849
64 168 320 520 768
104 152 200 248
48 48 48
That row is called the row of 3rd differences.
Now notice that they are all the same, so we don't need to
make any more rows of differences. So we assume that the next
3rd difference, if we had the next number after 1849, would also
be a 48, so we write another 48 on the bottom row.
(I'll color it red):
9 73 241 561 1081 1849
64 168 320 520 768
104 152 200 248
48 48 48 48
Then we work backwards by adding. Add the 48 to the 248, getting 296,
Write that to the right of the 248. (I'll color it red too).
9 73 241 561 1081 1849
64 168 320 520 768
104 152 200 248 296
48 48 48 48
Still working backward, Add the 296 to the 768, getting 1064,
Write that to the right of the 768. (I'll color it red too).
9 73 241 561 1081 1849
64 168 320 520 768 1064
104 152 200 248 296
48 48 48 48
One more step working backwards. Add 1064 to the 1849, getting
2913, write it at the end of the first row, and there is your answer!:
9 73 241 561 1081 1849 2913
64 168 320 520 768 1064
104 152 200 248 296
48 48 48 48
Answer: The next term is 2913
Edwin
Answer by terryriegel(1)  (Show Source):
You can put this solution on YOUR website! It can be any number you want. The solution arises from the fact that anything times zero is just zero and also adding zero to something doesn't change it. Now granted the formula will get a bit lengthy but it is a valid solution and it can be any number you want. Lets start with a simple sequence of three random numbers between (1-10) from random.org I got 6,9,2,_ whats next, Ok back to random.org its came up with 5. So our problem is to come up with a formula f(n) that produces f(1)=6, f(2)=9, f(3)=2, and f(4)=5
f(1)=6
f(2)=9
f(3)=2
f(4)=5
Lets initially choose f(n) = 6 f(1) = 6 check
f(2) = 6 fail so lets refine it f(n) = 6 + (n-1)(3/1) f(1) = 6 check
f(2) = 9 check
f(3) = 12 fail so we refine further f(n) = 6 + (n-1)(3/1) + (n-1)(n-2)(-10/2) f(1) = 6 check
f(2) = 9 check
f(3) = 2 check
f(4) = -15 fail so we refine further f(n) = 6 + (n-1)(3/1) + (n-1)(n-2)(-10/2) + (n-1)(n-2)(n-3)(10/3) f(1) = 6 check
f(2) = 9 check
f(3) = 2 check
f(4) = 5 check 
So to answer the original question using the same methods the formula looks like this... 
f(7) = 0 simplified... 
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