If the number of copies sold of a certain book in its second year of

Copies sold in 1st year: \(F\)
Copies sold in 2nd year: \(S\)
Copies sold in 3rd year: \(T\)

the number of copies sold of a certain book in its second year of publication was 1/6 the number of copies sold in its first year of publication
=> \(S=\frac{1}{6}F\)

the number of copies sold in its third year of publication was three times the number of copies sold in its second year
=> \(T=3S\)

the number of copies sold the first year was how many times the average (arithmetic mean) of the number of copies sold (this should say "sold per year") in its second and third years of publication?
=> \(F=k\times\frac{S+T}{2}\), \(k=?\)

Writing everything in terms of \(F\) makes \(F=k*\frac{S+T}{2}\) become:
=>\(F=k\times\frac{\frac{1}{6}F+3\times\frac{1}{6}F}{2}\)

=>\(F=\frac{k}{12}(4F)\)

=> \(k=3\)

Alternatively, and preferably, we could choose numbers that fit (as the problem involves only ratios/multiples with no fixed values).

Choosing \(F=6\) gives \(S=1\), and \(T=3\).

Then the average of the third and second year is 2, and \(F=3\times2\)