## Galois 2023 Question 2, CEMC UWaterloo

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#### Difficulty: 5

#### This problem is tagged with galois, galois23, highschool.

**(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 $x + 4$.

(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"

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