Hi Jerry,
A sequence is a list of numbers that may be generated by
a particular rule. For example: 1, 3, 5, 7, 9, ... is a sequence of
odd numbers produced by the rule 2n + 1, where n = 1, 2, 3, ...
.
A summation is the sum of the terms in the sequence. For
example: 1 + 3 + 5 + 7 + ... + 27 = sigma{n = 1, 13} (2n
+ 1}.
This is an example of a recursive definition:
x(1) = 1,
x(n) = n * x(n - 1).
In such a definition, it is necessary to have one or more
BASE case(s) as well as one or more REPEATED call(s).
Say we wish to compute x(5):
From the above definition, x(5) = 5 * x(4), which means
x(5) will call x(4).
Then x(4) = 4 * x(3), which means x(4) will call
x(3).
Then x(3) = 3 * x(2), which means x(3) will call x(2)
Then x(2) = 2 * x(1), which means x(2) will call x(1),
the base case.
Then x(1) = 1 and the process of repeated calls
stops.
Next, we go back to x(2) and compute it as 2 * 1 = 2.
Then we go back to x(3) and compute it as 3 * 2 = 6.
Then we go back to x(4) and compute it as 4 * 6 = 24.
Finally, we compute x(5) as 5 * 24 = 120.
Hope you are clearer now. Thanks!
P.S. I am unsure of what you meant by mathematical reasoning?
Could you say more, by referring to your notes? Thanks!
Cheers,
Wen Shih