Questions: Sequences and Series

Question 1

Is the sequence 2, 4, 6 the same as the sequence 6, 4, 2?

• A. Yes — they contain the same numbers
• B. No — the order is different, and in a sequence, order matters
• C. Yes — both have three numbers that are even
• D. It depends on the context

Answer: (B)

In a sequence, order is essential. 2, 4, 6 starts with 2 and increases. 6, 4, 2 starts with 6 and decreases. They have the same elements but in different positions, making them different sequences. This is unlike a set, where {2, 4, 6} and {6, 4, 2} would be the same. The distinction between sets and sequences is fundamental in mathematics.

Question 2

What is the 5th term of the sequence that starts at 10 and subtracts 2 each time?

• A. 2
• B. 4
• C. 0
• D. 6

Answer: (A)

The sequence is 10, 8, 6, 4, 2. The 1st term is 10, the 2nd is 8, the 3rd is 6, the 4th is 4, and the 5th is 2. You can also use the position rule: term = 10 - 2(position - 1) = 10 - 2(4) = 2 for the 5th term.

Question 3

A sequence is expected to generally follow a pattern or rule.

• T. True
• F. False

Answer: False

A sequence is any ordered list — the elements can follow a rule (like 2, 4, 6, 8) or not (like 3, 17, 1, 42). What makes it a sequence is that position matters, not that a rule exists. Of course, sequences WITH rules are far more useful and interesting — they are the ones we study in mathematics. But the concept of sequence (ordered list) is broader than the concept of pattern (predictable regularity).

Question 4

Why does order matter in a sequence but not in a set?

Model answer: A set is defined by membership — which elements are in it. The set {A, B, C} and {C, B, A} have the same members, so they are the same set. A sequence is defined by position — which element is first, second, third. The sequence A, B, C has A in position 1, while C, B, A has C in position 1 — they are different. Order matters in sequences because many real-world things depend on order: steps in a recipe, digits in a number, letters in a word (TAR is not RAT).

This distinction — between unordered collections (sets) and ordered collections (sequences) — is foundational in mathematics and computer science. Students who grasp this early will understand why the order of operations matters in arithmetic, why function inputs must be ordered, and why algorithms require specific step sequences.