Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.24

24. Find three different formulas or rules for the terms of a sequence a_n if the first three terms of this sequence are 2, 3, 6.

Solution: We consider the first three terms of a sequence 2,3,6.
The simplest form of this sequence is
a_n=2^{n}-(n-1)
For n=1,2,3,...
We can even use recursive formula for this sequence, that is,
a_1=2
a_2=3
a_n=a_{n-1}.a_{n-2}
For n \geq 3
We can also formulate the nth term of this sequence by inserting factorials and doing inspection.
2=\frac{(1+1)!}{2^0}
3=\frac{(2+1)!}{2^1}
6=\frac{(3+1)!}{2^2}
Therefore, we have for n=1,2,3,...
a_n=\frac{(n+1)!}{2^{n-1}}

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