If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3$, $5$, $6$ and $9$. The sum of these multiples is $23$. Find the sum of all the multiples of $3$ and $5$ below $1000$.

Here, we just take a number $n$ and a list of divisors then check if any of the divisors divides $n$.

-- | Check if any of the divisors divide the number n.
anyDivisor :: (Foldable c, Integral a) => a -> c a -> Bool
anyDivisor n divisors =
  any (\x -> n `mod` x == 0) divisors
  -- ^ NOTE: this short-circuits,
  --   i.e. stops as soon as it finds a single divisor.

Generate a list of all the numbers below $1000$ that are multiples of $3$ or $5$, then sum them.

solve1 :: (Foldable c, Integral a) => a -> c a -> a
solve1 bound divisors =
  sum $ filter (`anyDivisor` divisors) [1..bound-1]
  -- ^ sum the list of elements in the range [1..bound-1]
  --   that are divisible by any of the divisors

-- >>> solve1 1000 [3,5]
-- 233168

Haskell is a lazy language. Instead of explicitly generating the entire list of values, it keeps the specification at hand and generates the values on-demand, then discards each value as soon as it is no longer needed and it is efficient to discard it.

Approach 1 is only efficient because Haskell is lazy by design. Otherwise, generating a list of all values divisible by $3$ and $5$ when we are only interested in their sum is wasteful and would not scale to larger bounds. A generally better approach is to generate one number at an instance, add it to our running sum, generate the next, and so on. That way, we only need to keep memory for a single variable at a time.

However, most functional programming languages bar explicit iteration — much of iterative computing involves state/value mutation, and breaking functional purity. Most functional languages prefer recursion over iteration, since computing with explicit iteration usually involves mutating some state that lives across the iterations.

However, recursion is not a straight-up replacement for iteration. Take, for instance, a simple recursive solution to the factorial function:

def factorial(bound):
  acc = 1
  for i in range(1, bound+1):  # what happens if we start at 0?
    acc *= i                   # mutating the accumulator
  return acc

Here's a naive recursive solution in Haskell:

factorial :: Integer -> Integer
factorial n
  | n == 0    = 1
  | otherwise = n * factorial (n - 1)

What happens at runtime? First, recursive calls require a stack frame on each call, meaning each call to factorial allocates memory for the call's context, and that memory must not be freed until the call returns. In contrast to iteration, the recursive call must perform two iterations: one "forward" iteration setting up the nested recursive calls needed to compute the result, and one "backward" iteration to unwind the stack and compile the final result from the nested calls.

For example, a call to factorial 5 would result in the following stack trace.

factorial 5                                   -- call 1
5 * (factorial 4)                             -- call 2
5 * (4 * (factorial 3))                       -- call 3
5 * (4 * (3 * (factorial 2)))                 -- call 4
5 * (4 * (3 * (2 * (factorial 1))))           -- call 5
5 * (4 * (3 * (2 * (1 * (factorial 0)))))     -- call 6
5 * (4 * (3 * (2 * (1 * 1))))                 -- unwind 6
5 * (4 * (3 * (2 * 1)))                       -- unwind 5
5 * (4 * (3 * 2))                             -- unwind 4
5 * (4 * 6)                                   -- unwind 3
5 * 24                                        -- unwind 2
120                                           -- unwind 1 (final return)

The real problem is that memory allocated for any specific recursive call's stack frame in the forward iteration may not be freed until the backward iteration reaches the specific call and the specific recursive call completes execution and returns. We effectively use $\Theta (n)$ memory.

If we can do better, shouldn't we?

If a non-branching recursive call is the last computation in a function, why not pass the context forward and avoid setting up a new stack frame for the recursive call? This is the main idea behind tail recursion — eventually, the terminating call (i.e. base case) returns the solution back to the original caller.

def factorial(n: int, acc=1):
  if n == 0:
    return acc
  return factorial(n - 1, acc * n)

A similar solution in Haskell:

factorial :: Integer -> Integer
factorial n = iter n 1
  where
    iter :: Integer -> Integer -> Integer
    iter n acc
      | n == 0 = acc
      | otherwise = iter (n - 1) (acc * n)

Now, our stack trace is much simpler, and we only use $\Omega (1)$ memory:

factorial 5     -- call 
iter 5 1        -- subroutine call 1 
iter 4 5        -- subroutine call 2 
iter 3 20       -- subroutine call 3 
iter 2 60       -- subroutine call 4 
iter 1 120      -- subroutine call 5 
iter 0 120      -- subroutine call 6 
120             -- final return

Please remember to solve your original problem.

solve2 :: (Foldable c, Integral a) => a -> c a -> a
solve2 bound divisors = iter 0 0
  where
    iter acc curr
      | curr == bound             = acc
      | anyDivisor curr divisors  = iter (acc + curr) (curr + 1)
      | otherwise                 = iter acc (curr + 1)

-- >>> solve2 1000 [3,5]
-- 233168

With tail recursion, we effectively iterate (recursively!) through all values in the range and, when a value is a multiple of any of the divisors, we adjust¹ the value of the accumulator that is passed forward to the next recursive call.

Tail recursion emulates iteration with recursion. But can we explicitly iterate?

Yes, yes we can.

Albeit without some of the safety guarantees that functional programming gives us when our functions are pure.

solve3 :: (Foldable c, Integral a) => a -> c a -> a
solve3 bound divisors =
  runST $ newSTRef 0 >>= compute
  where
    compute total = do
      for [1..bound-1] $ \curr -> do
        when (curr `anyDivisor` divisors) $ do
          modifySTRef total (+ curr)
      readSTRef total

-- >>> solve3 1000 [3,5]
-- 233168

Instead of a tail recursion (which is unrolled into a loop by GHC) anyway, use the state monad to maintain the accumulator, and an explicit loop to check multiples and sum up values in the accumulator.