Question 643336
<font face="Times New Roman" size="+2">


Use the process of successive differences.


3 minus 0 is 3


7 minus 3 is 4


12 minus 7 is 5


18 minus 12 is 6


Then


4 minus 3 is 1


5 minus 4 is 1


6 minus 5 is 1


Since all of the differences are the same after two passes, you know a quadratic will suffice for a rule (one pass: linear, three passes: cubic, etc.)


So you need a function of the form *[tex \LARGE f(n)\ =\ an^2\ +\ bn\ +\ c]


Plugging in numbers you get


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(1)^2\ +\ b(1)\ +\ c\ =\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(2)^2\ +\ b(2)\ +\ c\ =\ 3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(3)^2\ +\ b(3)\ +\ c\ =\ 7]


Which leads to the following 3X3 system:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ =\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4a\ +\ 2b\ +\ c\ =\ 3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 9a\ +\ 3b\ +\ c\ =\ 7]


Which you can solve by your favorite process to get the coefficients of your function.


There is actually a shortcut for sequences.  The lead coefficient of your function is always one-half of the value of the value of the constant difference.  The constant difference here was 1, so the lead coefficient is 1/2.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>