(Galois 2023, Question 2, CEMC - UWaterloo)

In an Ing sequence, the first term is a positive integer and each term after the first is determined in the following way:

• if a term, $x$, is odd, the next term is $x + 3$, and
• if a term, $x$, is even, the next term is $x + 4$.

For example, if the first term in an Ing sequence is $13$, then the second term is $16$, and the third term is $20$.
(a) If the first term in an Ing sequence is $7$, what is the fifth term in the sequence?
(b) If the fifth term in an Ing sequence is $62$, what are the two possibilities for the first term?
(c) If the first term in an Ing sequence is $49$, determine the terms appearing in the sequence whose values are greater than $318$ and less than $330$.
(d) The number $18$ appears somewhere in an Ing sequence after the first term. If the first term is the positive integer $n$, determine all possible values of $n$.

Answer Submission Note(s)
In each of parts (b), (c), and (d), sort your answers in ascending order and separate them with a comma.
Separate the answers for each part with a single space.
For example: "a b1,b2 c1,c2 d1,d2"