3. Assessment

The following questions were given to Chat GPT.

Determine what's correct, fix all the mistakes, and assign a letter grade: A,B,C,D,F.

Rainbow sequences are a special flavor of infinite sequences.

They are defined inductively as follows.

i) The infinite sequence of increasing positive odd integers (1,3,5,7,9,11,...)

is a rainbow sequence.

ii) If (t₁, t2, t3,ti,) is a rainbow sequence, then (t₁, t2, t3,..., ti + 2,...)

(the sequence just like the first one, except any single value t, is replaced with

the value ti + 2) is a rainbow sequence.

iii) If (t₁, t₂, t3,.,t₁) is a rainbow sequence, then (t₁, t2, t3,..., t; — 2,...)

(the sequence just like the first one, except any single value ti is replaced with

the value ti-2) is a rainbow sequence.

a) Prove by structural induction that every rainbow sequence is an infinite se-

quence only consisting of odd values.

1. Base Case: The initial rainbow sequence, (1, 3, 5, 7, 9, ...), is an infinite sequence consisting

only of odd values. This is true because it consists of positive odd integers.

2. Inductive Step: Assume that a rainbow sequence, (t1, t2, t3,..., ti, ...), consists of only odd

values. We need to show that the sequence obtained by applying either rule ii) or rule iii) will

also consist of only odd values.

• Applying rule ii): The value ti + 2 is obtained by adding 2 to an odd integer ti. Since adding 2

to an odd integer always results in an odd integer, the new sequence will still consist of only

odd values.

• Applying rule iii): The value ti-2 is obtained by subtracting 2 from an odd integer ti. Again,

subtracting 2 from an odd integer always results in an odd integer, so the new sequence will

still consist of only odd values.

Therefore, based on the base case and the inductive step, we can conclude that every

rainbow sequence is an infinite sequence consisting only of odd values./nb) Show that it is not the case that all infinite sequences consisting of only odd

values are rainbow sequences. In other words, disprove the converse of part

b).

c) To disprove the converse of part b) and show that not all infinite sequences consisting

only of odd values are rainbow sequences, we can provide a counterexample.

Consider the infinite sequence (1, 3, 5, 7, 9, 11, ...). This sequence consists only of odd values,

but it does not follow the rules of rainbow sequences.

According to the rainbow sequence rules, the values in the sequence must be obtained by

either adding 2 or subtracting 2 from the previous values. In the provided sequence, the

difference between consecutive terms is always 2. However, in a rainbow sequence, the

difference between consecutive terms alternates between 2 and 0.

Therefore, the sequence (1, 3, 5, 7, 9, 11, ...) consisting only of odd values is not a rainbow

sequence, disproving the converse of part b).