SOLUTION: May I ask you a help for this question - Find he next term in the following sequence -3 2 1 0 5 22 57 116 Thank you

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Question 1136052: May I ask you a help for this question -
Find he next term in the following sequence
-3 2 1 0 5 22 57 116
Thank you
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

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first find differences:
-3+| | 2 | | 1 | | 0 | | 5 | | 22 | |+57 | | 116
| 5 | | -1 | |+-1 | | 5 | |+17 | |+35 | |+59 |
| | -6 | | 0 | | 6+| |+12 | |+18 | | 24 | |
| | | 6 | | 6 | | 6+| | 6 | |+6 | | |

you have three differences and you have cubic sequence
a%5Bn%5D+=+an%5E3+-+b+n%5E2+%2B+c+n+%2Bd

1th term is -3 and it is equal to a%2Bb%2Bc%2Bd
-3=a%2Bb%2Bc%2Bd
since third difference is 6, means 6a=6 =>a=1
1th term of second difference is -6, and it is equal to 12a%2B2b
12%2A1%2B2b=-6
2b=-6-12
2b=-18
b=-9

2nd term minus 1st term is 5, and it is equal to 7a%2B3b%2Bc
7a%2B3b%2Bc=5.........plug in a and b
7%2A1%2B3%28-9%29%2Bc=5
7-27%2Bc=5
-20%2Bc=5
c=5%2B20
c=25
go back to 1th term -3=a%2Bb%2Bc%2Bd
-3=1-9%2B25%2Bd
-3=17%2Bd
d=-3-17
d=-20
so, your formula is:
a%5Bn%5D+=+n%5E3+-+9+n%5E2+%2B+25+n+-+20
the next term in the following sequence is 9th term, means n=9
a%5B9%5D+=+9%5E3+-+9%2A9%5E2+%2B+25+%2A9+-+20
a%5B9%5D+=+205