8.1: Sequences

A sequence is an ordered list of entries, usually numbers, based on a repeating pattern.

Arithmetic sequences

The next entry in the sequence is the previous entry plus a fixed number.

• Example 1: 2, 6, 10, 14, …
• Pattern: Start at 2 and increase by steps of 4.
• What is the next (5th) entry in the sequence? 14 + 4 = 18.
• What would be the “zero-th” (0th) entry (the one before the first entry)? 2 – 4 = –2.
• What is a formula for finding the nth entry?
  • step x n + 0th entry
  • i.e., nth entry = 4n – 2.
• Check: 1st entry = 4(1) – 2 = 2.
• Check: 5th entry = 4(5) – 2 = 18.
• What is the 100th entry? 4(100) – 2 = 398.

• Example 2: 2, –3, –8, –13, …
• Pattern: Start at 2 and increase by steps of –5.
• What is the next (5th) entry in the sequence? –13 – 5 = –18.
• What would be the “zero-th” (0th) entry? 2 – (–5) = 7.
• What is a formula for finding the nth entry?
  • step x n + 0th entry
  • i.e., nth entry = –5n + 7.
• Check: 1st entry = –5(0) + 7 = 7.
• Check: 5th entry = –5(5) + 7 = 18.
• What is the 100th entry? –5(100) + 7 = –493.

Geometric sequences

The next entry in the sequence is the previous entry multiplied by a fixed number.

• Example 1: 3, 3.6, 4.32, 5.184, …
• Pattern: Start at 3 and increase by repeatedly multiplying by 1.2.
• What is the next (5th) entry in the sequence? 5.184 x 1.2 = 6.2208.
• What would be the “zero-th” (0th) entry (the one before the first entry)? 3 ÷ 1.2 = 2.5.
• What is a formula for finding the nth entry?
  • multiplierⁿ x 0th entry
  • i.e., nth entry = 1.2ⁿ x 2.5.
• Check: 1st entry = 1.2¹ x 2.5 = 3.
• Check: 5th entry = 1.2⁵ x 2.5 = 6.2208.
• What is the 100th entry? 1.2¹⁰⁰ x 2.5 = 207,044,936.

• Example 2: 57, 54.15, 51.4425, 48.870375, …
• Pattern: Start at 57 and decrease by repeatedly multiplying by 0.95.
• What is the next (5th) entry in the sequence? 48.870375 x 0.95 = 46.42685625.
• What would be the “zero-th” (0th) entry (the one before the first entry)? 57 ÷ 0.95 = 60.
• What is a formula for finding the nth entry?
  • multiplierⁿ x 0th entry
  • i.e., nth entry = 0.95ⁿ x 60.
• Check: 1st entry = 0.95¹ x 60 = 57.
• Check: 5th entry = 0.95⁵ x 60 = 46.42685625.
• What is the 100th entry? 0.95¹⁰⁰ x 60 = 0.35523175.

The video below works through some examples of sequences.