- Level: International Baccalaureate
- Subject: Maths
- Word count: 846
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 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:
n | ||
1 | 1 | 1 |
2 | 4 | 5 |
3 | 18 | 23 |
- | - | - |
- | - | - |
n | ! | ? |
The same results of from Table 1.
Middle
is true
- If is true, then
If k=k+1, then
Now,
=
=
=
=(k+2)!-1 (k+2)!
Thus is true whenever is true and is true.
is true for all n
Part 5:
!, I use this formula to show that is also true by simplifying to equal !
! = (n+1)!-n!
= (n+1)n!-n! (n+1)! = (n+1)n!
= n! (n+1-1)
= !
!
Now, I use to device a direct proof for the expression of that I conjectured in Part 3.
!
From Part 3, I know that
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:
The conjecture of is proven
Part 6:
Conclusion
Graph 2.1
Part 9:
The conjecture that I derived in Part 7 for can be proven through Mathematical Induction:
is:
+ (2+2)!-2! + … +
- If n=1
LHS:
= (1+2)!-1!
= 5
RHS: (1+1)!(1+3)= 5
is true
- If is true, then
+ (2+2)!-2! + … +
If k=k+1, then
Now,
=
=
= (k+1)!(k+2)
= (k+2)!(k+4)
Thus is true whenever is true and is true.
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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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