sequence Urgent!!!

In fact, to deal with the problems of sequence, you have to be observant. That is to observe is there any patterns in the sequence?


1. 0, 3, 8, 15, 24, ...

We can see that the sequence is in fact:

0, 3, 3 + 5, 8 + 7, 15 + 9, ...

The next term in the sequence is in fact the sum of the previous term and a number, with arithmetic sequence.

So, the sequence can be expressed in:

an = an-1 + [3 + 2(n - 2)]

an = an-1 + (2n - 1), where n >= 2 and first term, a1 = 0


2. -9, 101, -999, 10 001, -99 999, ...

To be observant, the sequence is as follows:

-(10 - 1), 100 + 1, -(1000 - 1), 10 000 + 1, -(100 000 - 1)

To be more precise, it is:

(-1)[10 + (-1)], (-1)2[(10)2 + (-1)2], (-1)3[(10)3 + (-1)3], (-1)4[(10)4 + (-1)4], (-1)5[(10)5 + (-1)5], ...

Therefore, the sequence is:

an = (-1)n[(10)n + (-1)n], where n is a positive integer.






2009-03-26 09:27:53 補充：
For example 1, just observe the terms, it is the sum of the previous term and a number, which is 3, 5, 7, 9, ... with a pattern that the next number exceeds that of the previous one by 2. So, we have the nth term, an = an-1 + (2n - 1)
where an-1 is the previous term.

2009-03-26 09:28:56 補充：
For example, if n = 4, that is the fourth term, it is 15
We put n = 4 into the equation, then a3 is the previous term, a3 = 8
And L.H.S. = 15
R.H.S. = 8 + 2(4) - 1 = 15 = L.H.S
The result is verified.