As Elsa says, it's when the 'to' is a preposition - belonging to the previous phrase - rather than part of the infinitive. Here are some examples of phrases ending in 'to' which need the following verb to be in the gerund (+ ing) form: The most common ones are: look forward to ( "I'm looking forward to seeing you") be used to ("He's not used to driving on the left") There are also these ones: confess to admit to adjust to be accustomed to be devoted to be addicted to be committed to be dedicated to

This is only possible when "to" functions as a preposition, not as part of the infinitive "TO DO." In your examples, "look forward to" and "an approach to" TO is a preosition and therefore followed by either an object or a gerund. Hope that helps!

Let me think for a while. I am on my way to doing research into the usage of "to doing." I admitted to doing this bad thing. I hope my admission won't lead me to doing more harm than good.

That's such a great question, so I've just posted an in-depth explanation as a discussion topic: https://www.italki.com/discussion/164168

Thank you for answering my questions. I appreciate it