22nd term of the A.P.: $\frac{3}{2}, \frac{1}{2}, \frac{-1}{2}, \frac{-3}{2}, ........ $ is

22nd term of the A.P.: $\frac{3}{2}, \frac{1}{2}, \frac{-1}{2}, \frac{-3}{2}, ........ $ is

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} $.

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