Find the next 3 terms of the sequence

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Question from Lanelyn, a student:

Find the next 3 terms of the sequence 2,3,9,23,48,87,__,__,__

One way that sometimes works is the "method of finite differences."

Write the numbers out. Call this "row 0"
Beneath each pair write their difference. Call this "row 1". Use a minus sign if the second is smaller than the first.
Beneath each pair in row 1 write their differences - call this "row 2"

Repeat till you see a pattern that you can extend. Extend it and work back up, adding.

EXAMPLE: 1,3,7,13,21,31,43,_,_,_,

1		3		7		13		21		31		43
	2		4		6		8		10		12
		2		2		2		2		2

1		3		7		13		21		31		43
	2		4		6		8		10		12
		2		2		2		2		2		2

12+2=14, 14+2=16, ...
43+14 = 57, (etc)

1		3		7		13		21		31		43		57		73		91
	2		4		6		8		10		12		14		16		18
		2		2		2		2		2		2		2		2

If you get all numbers the same in row d, your numbers are successive values P(0), P(1), P(2),... of a polynomial of degree d. There are ways of finding the coefficients of this polynomial.

Good Hunting!
RD

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