Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.23

23. Find three different formulas or rules for the terms of a sequence {a_n}  if the first three terms of this sequence are  1,2,4.

Solution: We consider the first three terms of a sequence as 1,2,4

The simplest form of this sequence is

a_n=2^{n-1}
For n=1,2,3,...
We can even use recursive formula for this sequence, that is,
a_1=1
a_2=2
a_n=a_{n-1}+2a_{n-2}
For n \geq 3
We can also formulate the nth term of this sequence by inspection.
1=\frac{1^2-1+2}{2}
2=\frac{2^2-2+2}{2}
4=\frac{3^2-3+2}{2}
Therefore, we have for n=1,2,3,...
a_n=\frac{n^2-n+2}{2}

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