Given A.P.: $ \displaystyle \frac{3}{2}, \frac{1}{2}, \frac{-1}{2}, \frac{-3}{2}, \dots $
First term: $ \displaystyle a = \frac{3}{2} $
Common difference:
$ \displaystyle d = \frac{1}{2} - \frac{3}{2} = -1 $
The general formula for the $ \displaystyle n $th term of an A.P. is:
$ \displaystyle a_n = a + (n-1) d $
Substituting $ \displaystyle n = 22 $:
$ \displaystyle a_{22} = \frac{3}{2} + (22-1)(-1) $
$ \displaystyle = \frac{3}{2} + 21(-1) $
$ \displaystyle = \frac{3}{2} - 21 $
$ \displaystyle = \frac{3}{2} - \frac{42}{2} $
$ \displaystyle = \frac{-39}{2} $
Thus, the 22nd term is $ \displaystyle \frac{-39}{2} $.