Nth Term of a Sequence in Haskell, Haskell

Language: Layout:

module Main (
  nth_term,
  main
) where

all_equal :: (Eq e) => [e] -> e -> Bool
all_equal [] _ = False
all_equal [el] x = el == x
all_equal (el:ys) x = el == x && all_equal ys x

all_same :: (Eq e) => [e] -> Bool
all_same [] = False
all_same li@(x:xs) = all_equal li x

get_steps :: (Num a) => [a] -> [a]
get_steps xs = [xs !! (n + 1) - xs !! n | n <- [0..length xs - 2]]

nth_term :: (Num a, Ord a, Show a, Fractional a, Floating a) => [a] -> String
nth_term xs =
  if all_same xs then
    show $ head xs
  else
    if all_same steps then
      let
        step = head steps
        add = head xs - step
        
        d_step = if step == 1 then "" else show step
        d_add = if add == 0 then ""
          else if add > 0 then '+':(show add) else show add
      in d_step ++ "n" ++ d_add
    else
      let
        n_steps = get_steps steps
      in
        if all_same n_steps then
          let
            n_step = head n_steps
            coefficient = n_step * 0.5
            normal = [(fromIntegral n ** 2) * coefficient | n <- [1..length xs - 1]]
            difference = [j - k | (j, k) <- zip xs normal]
            nth_diff = nth_term difference
            
            d_coefficient = if coefficient == 1 then "" else show coefficient
            d_nth_diff = if head nth_diff == '0' then "" else (if head nth_diff == '-' then
              "" else "+") ++ nth_diff
          in d_coefficient ++ "n²" ++ d_nth_diff
        else "Invalid"
  where
    steps = get_steps xs
      
main = do
  putStrLn $ nth_term [3, 9, 19]

Show compiler warnings [ + ] Compiler args [ - ] Show input
Compilation time: 0.62 sec, absolute running time: 0.14 sec, cpu time: 0 sec, memory peak: 3 Mb, absolute service time: 0,77 sec