# 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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