2.3.1 Generate sequence term-to-term / position-to-term rule

• In an arithmetic sequence, each term increases by a constant difference.


  True
• Steps to generate a sequence using a term-to-term rule:

  1️⃣ Identify the first term

  2️⃣ Apply the rule to get the next term

  3️⃣ Repeat the process to generate additional terms
• Match the term position with its value in the example sequence:

  1 ↔️ 4

  2 ↔️ 6

  3 ↔️ 8

  4 ↔️ 10
• What is the first term in the example sequence?

  4
• Steps to generate terms using the rule \(3n - 1\)

  1️⃣ Substitute \(n = 1\)

  2️⃣ Calculate the term value: \(3(1) - 1 = 2\)

  3️⃣ Substitute \(n = 2\)

  4️⃣ Calculate the term value: \(3(2) - 1 = 5\)

  5️⃣ Continue for subsequent positions
• What does a term-to-term rule describe in a sequence?

  How to get next term
• Match the position with its corresponding term in the sequence 2, 4, 6, 8...:

  1 ↔️ 2

  2 ↔️ 4

  3 ↔️ 6

  4 ↔️ 8
• What does understanding the pattern in a sequence help us predict?

  Future terms
• What is the second term in the example sequence?

  6
• What is a position-to-term rule?

  A formula to find terms
• What is the term at position 1 in the sequence 2, 4, 6, 8...?

  2
• Match the type of sequence with its example:

  Arithmetic sequence ↔️ 1, 3, 5, 7...

  Geometric sequence ↔️ 2, 4, 8, 16...
• Position-to-term rules allow you to directly find any term in a sequence without starting from the first term.


  True
• The example sequence is: 4, 6, 8, 10, 12...


  True
• What are the numbers in a sequence called?

  Terms
• What is the term value when the position \(n = 4\)?

  11
• What is the term-to-term rule for the sequence 5, 10, 20, 40...?

  Multiply by 2
• The term-to-term rule for the sequence 3, 6, 9, 12... is "Add 3 to the previous term".

  True
• How is the term value calculated for position 1 using the rule \(3n - 1\)?

  \(3(1) - 1 = 2\)
• What is the term value when the position \(n = 1\)?

  2
• What is the term at position 3 in the sequence 2, 4, 6, 8...?

  6
• Position-to-term rules require starting from the first term of a sequence.

  False
• Steps for checking a sequence using the term-to-term method

  1️⃣ Compare each term to the previous term using the term-to-term rule

  2️⃣ Ensure the rule consistently holds for all terms
• The position-to-term rule for the sequence 2, 5, 8, 11... is \( 3n - 1 \).


  True
• What is the formula used in the table to generate the term values?

  \(3n - 1\)
• What is the primary difference between a recursive and an explicit formula for sequences?

  Dependence on previous terms
• Checking sequences ensures accuracy and deepens understanding of the pattern.

  True
• Give an example of a recursive formula for a sequence.

  $a_{n} =$$a_{n - 1} +$$3$
• The term value for position 4 using the rule \(3n - 1\) is 11


  True
• The sequence 1, 4, 7, 10... can be defined by both a recursive and an explicit formula.

  True
• How is the nth term calculated using a recursive formula?

  Iterating through prior terms
• What is the explicit formula for the sequence 1, 4, 7, 10...?

  $a_{n} =$$3n - 2$
• Explicit formulas depend on previous terms to calculate the next term in a sequence.

  False
• The term value for position 3 using the rule \(3n - 1\) is 9


  False
• The term value for position 2 using the rule \(3n - 1\) is 5


  True
• What is the term value when the position \(n = 3\)?


  8