Are both examples correct or is there only one way to write it?
The program is for people from the age of fifteen to twenty-one.
OR
The program is for people from the ages of fifteen to twenty-one.
To me ages (plural) makes sense because I am talking about more than one age but I am not sure if it's correct.
In your example, only the singular is possible. A range of ages ("from age 1 to age 2") starts at one age (singular) and ends at another. We can also talk about the period between a pair of ages (plural).
The expression "from the age of fifteen to the age of twenty-one" can be written without the second occurrence of "from the age of", as in your example.
During the period from the age of twenty to that of thirty, the mortality was eight per cent.; from thirty to forty, thirteen and one- tenth ; from forty to fifty, thirteen
From the age of twenty to the age of forty, Bessemer lived the life of a journeyman inventor.
The law also imposed penalties on orbi, that is, married persons who had no children (qui liberos non habent, Gaius, ii.111) from the age of twenty-five to sixty in a man, and from the age of twenty to fifty in a woman.
Every Frenchman therefore is a member of the army practically or potentially from the age of twenty to the age of forty-five.
A man from the age of twenty to sixty shall pay twenty-five dollars;
It took me from the age of twenty to twenty six to finally realize I could be a good husband to the woman I loved
Way back in 1908, Kreisler (in an interview) said in the October issue of The Strad 'I am thirty-three now, and from the age of twenty to twenty-seven I struggled
Sunglasses are one of the only pieces in a gentleman's wardrobe that he can have from the age of twenty to sixty without having to change
