I mentioned in my last email that you can do that with extra type parameter.
Apparently, it’s much simpler but my code needs some improvement:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
​
{-# LANGUAGE RankNTypes #-}

newtype Constant a = Constant a deriving (Eq, Ord, Num, Enum)

data Linear    a = Linear    a (Constant a)
data Quadratic a = Quadratic a (Linear a)
data Arbitrary a = Arbitrary (forall n. Integral n => n -> a)

class PartialSum term where
  partialSum :: (Integral a, Num b) => a -> term b -> b

instance PartialSum Constant where
  partialSum n (Constant c) = fromIntegral n * c

instance PartialSum Linear where
  partialSum n (Linear k c) = k * fromIntegral (n * (n + 1) `div` 2) + partialSum n c

instance PartialSum Quadratic where
  partialSum n (Quadratic k l) = k * fromIntegral (n * (n + 1) * (2 * n + 1) `div` 6)  + partialSum n l

instance PartialSum Arbitrary where
  partialSum n (Arbitrary f) = sum $ map f [1..n]

This gives you appropriate behavior:

-- integers
partialSum 10 $ Linear 1 0
partialSum 10 $ Arbitrary (2^)
-- floats
partialSum 10 $ Arbitrary (\n -> 1.5 * fromIntegral n :: Float)
partialSum 10 $ Arbitrary (sqrt . fromIntegral)
-- complex (from Data.Complex)
partialSum 10 $ Arbitrary (\n -> fromIntegral n^2 :+ fromIntegral n)

Nick