arithmetic progression, finding the nth term. The sum of the 1st n terms, of an AP is $S_n=n^2-3n$. Write down the 4th term and find an expression for the $n$th term.

Will the 4th term be $t_4= a+3d$?
Let the arithmetic sequence be 

$ a, a + d, a + 2d, \cdots $

The sum of the first $ n $ term is 

$ a + (a + d) + \cdots + a + (n - 1)d $

Remember the sum is simply the (first term + the last term) * number of terms divide by 2, so we have

$ n^2 - 3n = S_n = \frac{(a + a + (n - 1)d)n}{2} = \frac{(2a - d)n + n^2d}{2} $

So match coefficients we can get $ \frac{d}{2} = 1, \frac{2a - d}{2} = -3 $

Now it is obvious that $ d = 2 $, $ a = -2 $.

The 4th term is $ a + 3d = 4 $, the nth term is $ a + (n - 1)d = 2n - 4 $.
Obviously,
$$t_4=4^2-3\cdot4=4.$$
Then
$$S_1=a=1^2-3\cdot1=-2$$ and $$S_2=a+(a+d)=2^2-3\cdot2=-2$$ give you $a$ and $d$.