is the article 'the' needed in the second noun in this sentence:
Keith forgot to turn off the television and (the) fan.
Thanks.
Yes, two articles are needed as they are denoting two different objects - the television and the fan.
Let me illustrate this with an example.
The police was chasing the traitor and the thief.
In the above sentence, the police was chasing two different people - one was a traitor and the other was a thief. Now see the following sentence:
The police was chasing the traitor and thief.
In this sentence, the police was chasing only one person who was both a traitor and thief.
So, the use of two articles is necessary when you are talking about two different things or people.
The short answer is: it depends on the context. Use articles if it improves the understanding of the sentence. They are often not needed if the context is clear.
Expanding on your example:
Keith forgot to turn off the television, fan, oven, and lights.
vs.
Keith forgot to turn off the television, the fan, the oven, and the lights.
The first version with only one article is correct because there can be no reasonable confusion about what you are saying. The second version is also correct and may be used for emphasis.
In situations where the nouns could be the same, articles can help indicate that they are different. satnam gave a good example where omitting an article could mean one or two people:
The police were chasing the traitor and thief.
This gets even more confusing because sometimes an article will be used with both nouns for emphasis while referring to one thing:
Keith turned off the television and the entertainment. With nothing else to do, his friends went home.
In this case, the television is the entertainment.
So, again, context will decide whether you need to use articles. It is largely preference.