**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}