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Investigating a sequence of numbers

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Introduction

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 image00.pngis defined by

image01.png, image66.png, image75.png, …

image40.pngFrom the pattern of different values of n in image83.png above, I conclude thatimage92.png!

Part 2:

Letimage102.png

        If n=1

image100.png = image112.pngwere image02.png!

image13.png!

image21.png

        If n=2

image29.pngwere image02.png!, image42.png!

image52.png

image61.png+ (2 image64.png

image65.png

image67.png

        If n=3

image68.pngwere image02.png!, image42.png!,image69.png!

image70.png

image71.png+ (2 image72.png

image73.png

image74.png

Part 3:

From Part 2, I know that:

image76.png!

To conjecture an expression of image39.png, I first organize the results that are derived in Part 2 to discover a pattern in the value of image39.pngas n increases.

Table 1.1:

n

image03.png

image39.png

1

1

1

2

4

5

3

18

23

-

-

-

-

-

-

n

image77.png!

?

The same results of image78.png from Table 1.

...read more.

Middle

image87.png is true

  1. If image88.png is true, then

image89.png

        If k=k+1, then

image90.png

        Now, image91.pngimage93.png

                      = image94.png

                      =image95.png

                      =image96.png

                      =(k+2)!-1   image58.pngimage97.png (k+2)!

Thus image98.pngis true whenever image99.pngis true and image100.png is true.

image101.png is true for all n

Part 5:

image58.pngimage92.png!, I use this formula to show that image103.png is also true by simplifying image104.png to equal image105.png!

image105.png! = (n+1)!-n!

                   = (n+1)n!-n! image58.png (n+1)! = (n+1)n!

                   = n! (n+1-1)

                   = image105.png!

image40.pngimage106.png!

 Now, I use image103.png to device a direct proof for the expression of image81.pngthat I conjectured in Part 3.

image58.pngimage106.png!

image107.png

image108.png

image109.png

From Part 3, I know thatimage110.png

image40.pngimage111.png

When the first value of the first term is subtracted from the second value of the following term, 0 is derived so I cancel these terms. After I cancel the values to the most simplified manner, -1! and (n+1)! are left in the expression from which the following equation is derived:

image113.png

image40.pngThe conjecture of image81.pngis proven

Part 6:

...read more.

Conclusion

. From the two graphs, I notice that (n+1)!(n+3) is exactly greater by 3 units to image31.png for all three points image40.png I conclude that:image39.png = (n+1)!(n+3) -3image38.png

Graph 2.1

Part 9:

The conjecture that I derived in Part 7 for image27.png can be proven through Mathematical Induction:

image31.png is: image41.png

image43.png + (2+2)!-2! + … + image44.png

  1. If n=1

         LHS: image45.png

             = (1+2)!-1!

             = 5

                     RHS: (1+1)!(1+3)image33.png= 5

image46.png

image47.png is true

  1. If image48.png is true, then

image49.png

image50.png + (2+2)!-2! + … + image51.png

        If k=k+1, then

image53.png

        Now, image54.png

                =image55.png

                =image56.png

                = (k+1)!(k+2) image57.png

                = (k+2)!(k+4)image33.pngimage58.pngimage59.png

Thus image60.pngis true whenever image48.pngis true and image62.png is true.

image63.png is true for all n

Conclusion:

Through this investigation, I have developed my knowledge about series and sequences involving permutations. I have learnt to use the patterns in a series to conjecture an expression for it and I had an opportunity to utilize my awareness of mathematical induction into proving the general term for the series. Most importantly, I have learnt to use technology related to series involving permutations. I enjoyed this investigation.

image17.png

...read more.

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