An increasingly/increasing large number of people are using online travel sites rather than travel agents.
Increasingly is a adverb , increasing is adjective.
So Which should I choose?
Increasingly is a adverb , increasing is adjective.
Adverbs modify verbs or other modifiers.
Adjectives modify nouns.
So you have to figure out what increasing[ly] is modifying.
An increasingly/increasing large number of people are using online travel sites rather than travel agents.
Large is a modifier that is modifying people - and we are saying that it's getting larger and larger over time. The largeness is increasing.
So increasingly - the adverb is used.
Since the word increasing is modifying the words "large number", which is a noun, the correct word is increasing.
EITHER
An increasingly large number of people...
OR
An increasing number of people....
The second is more economical, but some people might think that the longer form adds emphasis.