Why does it work? “multiply(arr, n) == multiply(arr, n – 1) * arr[n – 1]”

I just can’t understand the lesson “Replace loops using recursion” of freeCodeCamp. I’ll quote that part below;

Recursion is the concept that a function can be expressed in terms of itself. To help understand this, start by thinking about the following task: multiply the first n elements of an array to create the product of those elements. Using a for loop, you could do this:

  function multiply(arr, n) {
    var product = 1;
    for (var i = 0; i < n; i++) {
        product *= arr[i];
    }
    return product;
  }

However, notice that multiply(arr, n) == multiply(arr, n – 1) * arr[n – 1] . That means you can rewrite multiply in terms of itself and never need to use a loop.

  function multiply(arr, n) {
    if (n <= 0) {
      return 1;
    } else {
      return multiply(arr, n - 1) * arr[n - 1];
    }
  }

Especially this part. multiply(arr, n) == multiply(arr, n – 1) * arr[n – 1]

I can understand if it’s like this;

multiply(arr, n) == multiply(arr, n – 1) * arr[n]

That’s because if arr = [2,3,4,5,6,7,8,9],

multiply(arr, 5); equals 2*3*4*5*6*7

multiply(arr, 4); equals 2*3*4*5*6

multiply(arr, 4) * arr[5]; equals (2*3*4*5*6)*7

So multiply(arr, n) and multiply(arr, n - 1) * arr[n] is the same value”

However, I can’t understand why multiply(arr, n) == multiply(arr, n – 1) * arr[n – 1] ? Can anyone please tell what’s happening in this code? Why they are equal?

Answer

However, I can't understand why multiply(arr, n) == multiply(arr, n - 1) * arr[n - 1] ? Can anyone please tell what’s happening in this code? Why they are equal?

The given algorithm is multiplying the first n elements of an array arr and returning the answer.

Now, to multiply the first n elements, we can multiply the first (n-1) elements and then multiply the result with the nth element of the array.

So, multiply(arr, n) == multiply(arr, n - 1) * arr[n - 1].

multiply(arr, n) means multiply the first n elements of the array arr.
multiply(arr, n - 1) means multiply the first n-1 elements of the array arr.