Posted by Alicia on Thursday, April 14, 2011 at 5:37am.
n &isin N, so we are dealing with integers.
This is a recurrence relation in which the nth term is defined as a function of the previous term, i.e. (n-1)th term.
Take the Fibonacci sequence, which is defined as
f(n)=f(n-1)+f(n-2)
f(0)=0
f(1)=1
for n∈N.
So f(2)=f(1)+f(0)=0+1=1
f(3)=f(2)+f(1)=1+1=2
f(4)=f(3)+f(2)=1+2=3
....
So to calculate f(7) using the above definition, you will need to calculate f(5) and f(4), which in turn will require you to calculate f(3) and f(2)...
Most recurrence relations have a closed form with which we can calculate the nth term without knowing the previous terms. However, without the knowledge of the closed form, in order to calculate the nth term, we need to calculate all the previous terms, namely 0th, 1st, 2nd, 3rd, ... (n-2)th, (n-1)th etc.
In this particular problem, you can only calculate up to n=7 where an is defined.
Given a(n)=n*a(n-1), and given a(1)=1.
So
a(2)=2a(1)=2*1=2
a(3)=3a(2)=3*2=6
...