Investigating a sequence of numbers

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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:



        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.

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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 ...

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