Quandaries
and Queries 

Jack year 10 student Hi I have a problem generated by sequences. If i had a sequence which was the same at the 4th difference, aka difference to the 4th, like: 1 5 14 30 56 96 (sequence) 4 9 16 26 40 5 7 10 14 2 3 4 1 1 then what would be the set formulars to find out a, b, c, d, and e as i know the equation has to be an^{4}+bn^{3}+cn^{2}+dn+e i mean this in the sence of with a cubic sequence how the 3rd and final difference always equals 6a, the first 2nd difference equals 12a + 2b, and so on, like: a+b+c+d 7a+3b+c 12a+2b 6a so whats the equations for differences to the 4th, i hope thats well put enough the question. Please Help! 

Hi Jack, If the sequence {1,5,14,30,....} is {A_{n}} for n≥0 and A_{n} = an^{4}+bn^{3}+cn^{2}+dn+e you can always write out 5 equations from the first 5 terms by setting n=0,1,2,3,4 and solving:
This is a bit of work but it will work out. Sometimes it is easy to see what the earlier differences are by inspection and you can save some arithmetic but this isn't in general a good attack. Penny 

