# Investigating a sequence of numbers

Type 1: Investigating a sequence of Numbers

This is an investigation about series and sequences involving permutations. From a given series, I find the pattern of numbers that result from different values and use graphs to conjecture an expression from the series. By using mathematical induction and direct proof, I prove the general terms that I derived for the series.

Part 1:

The sequence of numbers is defined by

, , , …

From the pattern of different values of n in  above, I conclude that!

Part 2:

Let

If n=1

= were !

!

If n=2

were !, !

+ (2

If n=3

were !, !,!

+ (2

Part 3:

From Part 2, I know that:

!

To conjecture an expression of , I first organize the results that are derived in Part 2 to discover a pattern in the value of as n increases.

Table 1.1:

The same results of  from Table 1.1 can be represented as follows:

Table 1.2 :

From the patterns exhibited in Table 1.2, I notice that which is further illustrated in Graph 1.1.

In Graph 1.1, I plotted the graph of for the first three values (represented by green dots) and I assumed that (n+1)! will lead to a conjecture forand plotted its values for n=1,2,3 (represented by red dots) . From the two graphs, I notice that (n+1)! is exactly 1 unit above  for all three ...