2.4 Sequences

• An example of a sequence defined by listing terms is 2, 4, 6, 8, ... where each term increases by 2
• What is a recurrence relation in the context of sequences?

  Defines terms based on previous terms
• In the recurrence relation $a_{n} =$$a_{n - 1} +$$2$, the initial value $a_{1}$ is 2
• A general formula eliminates the need to refer to previous terms in a sequence.

  True
• What is the general formula for the sequence 2, 4, 6, 8, ...?

  a_{n} = 2n</latex>
• In a geometric sequence, terms are multiplied by a constant common ratio.

  True
• What is the first step to identify the common difference in an arithmetic sequence?

  Find consecutive terms
• In a geometric sequence, the common ratio must be constant for all term pairs.

  True
• In the nth term formula for arithmetic sequences, $n$ represents the position
• A general formula for a sequence allows you to calculate any term directly based on its position
• What is the common factor in a geometric sequence?

  Common ratio
• What does it mean to verify consistency when identifying the common difference in an arithmetic sequence?

  Ensure it remains constant
• The common difference in an arithmetic sequence is denoted by $d$.


  True
• The general formula for an arithmetic sequence is a_{n} = a_{1} + (n - 1)d
• Match the sequence type with its common factor and general formula:

  Arithmetic ↔️ Common difference $d$, $a_{n} =$$a_{1} +$$(n - 1)d$

  Geometric ↔️ Common ratio $r$, $a_{n} =$$a_{1} \cdot r^{n - 1}$
• The common difference or common ratio must be verified for multiple term pairs to ensure consistency.

  True
• In an arithmetic sequence, you calculate the common difference by subtracting the earlier term from the later
• To use the nth term formula for arithmetic sequences, you must identify the first term and the common difference
• What is a recurrence relation used for in sequences?

  Defining terms based on previous ones
• In arithmetic sequences, terms increase by a constant common difference.

  True
• What is a sequence in mathematics?

  Ordered list of numbers
• A sequence can be defined by listing its terms directly.

  True
• The common ratio in a geometric sequence is calculated by dividing the later term by the earlier term.


  True
• In the arithmetic sequence 2, 5, 8, 11..., the common difference is 3
• In the nth term formula for arithmetic sequences, $a_{1}$ represents the first term.


  True
• Find the 10th term of the geometric sequence 2, 4, 8, 16, ...

  1024
• In a geometric sequence, $r$ represents the common ratio.
• The formula for the nth term of an arithmetic sequence is $a_{n} =$$a_{1} +$$(n - 1)$d.
• Steps to find specific terms in sequences

  1️⃣ Identify the first term and common difference/ratio

  2️⃣ Determine the term number

  3️⃣ Substitute values into the appropriate formula

  4️⃣ Calculate the value of $a_{n}$
• Arithmetic sequences have a constant common difference between consecutive terms.
• What is a sequence in mathematics?

  An ordered list of terms
• What is the key difference between arithmetic and geometric sequences?

  Common difference vs ratio
• What is the first step to identify the common difference in an arithmetic sequence?

  Choose consecutive terms
• Steps to identify the common ratio in a geometric sequence

  1️⃣ Choose two consecutive terms

  2️⃣ Divide the later term by the earlier term

  3️⃣ Verify consistency

  4️⃣ State the common ratio
• Match the sequence type with the correct identifying value:

  Arithmetic ↔️ $d$

  Geometric ↔️ $r$
• The nth term formula for arithmetic sequences is $a_{n} =$$a_{1} +$$(n - 1)d$.


  True
• What is the first step to use the arithmetic sequence nth term formula?

  Identify $a_{1}$ and $d$
• Steps to use the geometric sequence nth term formula

  1️⃣ Identify $a_{1}$ and $r$

  2️⃣ Determine the term position $(n)$

  3️⃣ Substitute values into the formula

  4️⃣ Calculate the value of $a_{n}$
• In the geometric sequence nth term formula, the value $a_{1}$ represents the first term
• What does the term $d$ represent in the formula for arithmetic sequences?


  Common difference