Here's a sequence:

2, 4, 8, 13, 20, 29, _, _.

What numbers go into the blanks, and what is the rule for the sequence?

This is just to clarify ohad's solution, since it's not exactly clear, but I think it was what Lewis was looking for..

First, consider a function:
f(n+1) = f(n) - n
Where f(0)=2, this gives the sequence:
2, 2, 1, -1, -4, -8, -13

Now, consider a sequence, for which S₀=2, and
S_n+1 = S_n + S_n-1 + f(n)

Now, we have:

n    f(n)    Sn


0      2            2


1      2            4 = 2 + 2


2      1            8 = 2 + 4 + 2


3     -1           13 = 4 + 8 + 1


4     -4           20 = 8 + 13 - 1


5     -8           29 = 13 + 20 - 4


6    -13           41 = 20 + 29 - 8


7    -19           57 = 29 + 41 - 13


8    -26           79 = 41 + 57 - 19

So, the numbers in the blanks should be 41 and 57.

Posted by DJ on 2003-07-29 12:56:08