3.4 Sequences and Series

• What is a sequence in mathematics?

  Ordered list of numbers
• What is the formula for the nth term of a geometric sequence?

  $a_{n} =$$a_{1} *$$r^(n - 1)$
• Match the type of sequence with its constant:

  Arithmetic Sequence ↔️ Common Difference

  Geometric Sequence ↔️ Common Ratio
• The common ratio in a geometric sequence is found by dividing any term by the term that precedes
• What does $a_{1}$ represent in the geometric sequence formula?


  The first term
• What is the formula to find the nth term of a geometric sequence?

  $a_{n} =$$a_{1} *$$r^{(n - 1)}$
• An arithmetic sequence is a list of numbers where each term increases or decreases by a constant difference
• The key difference between arithmetic and geometric sequences lies in how the terms change.


  True
• In the geometric sequence 3, 6, 12, 24, ..., the 5th term is 48
• The sum of the first $n$ terms of a geometric series is given by $S_{n} =$$\frac{a_{1}(1 - r^{n})}{1 - r}$, where $r \neq 1$ and $a_{1}$ is the first term
• The formula for the partial sum of an arithmetic series is $S_{n} =$$\frac{n}{2}(a_{1} +$$a_{n})$, where $a_{n}$ is the nth
• In the arithmetic series partial sum formula, 'n' represents the number of terms to sum.


  True
• What is a partial sum of a geometric series?

  Sum of specific terms
• What does $r$ represent in the geometric series partial sum formula?


  Common ratio
• For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1.


  True
• Match the sequence type with its formula:

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

  Geometric ↔️ $a_{n} =$$a_{1} \times r^{(n - 1)}$
• What is the constant factor between terms in a geometric sequence called?

  Common ratio
• What is the formula to find the nth term of an arithmetic sequence?

  $a_{n} =$$a_{1} +$$(n - 1)d$
• The nth term of a geometric sequence is found using the formula a_{n} = a_{1} \times r^{(n - 1)}</latex>.
• The sum of an arithmetic series is found using the formula S_{n} = \frac{n}{2}(2a_{1} + (n - 1)d)</latex>.
• What is the formula for the partial sum of an arithmetic series?

  $S_{n} =$$\frac{n}{2}(2a_{1} +$$(n - 1)d)$
• The partial sum of an arithmetic series is the sum of its first n terms.
• What is the formula for the partial sum of a geometric series?

  $S_{n} =$$\frac{a_{1}(1 - r^{n})}{1 - r}$
• An infinite geometric series converges if the absolute value of the common ratio is greater than or equal to 1.

  False
• Geometric sequences are used to model savings growth with annual interest.

  True
• Geometric sequences increase or decrease by a constant ratio
• The terms in an arithmetic sequence change by a constant difference
• The common difference in an arithmetic sequence is found by subtracting any term from the term that follows
• Steps to find the 5th term of a geometric sequence with $a_{1} =$$2$ and $r =$$3$ in the correct order:


  1️⃣ Identify $a_{1}$, $r$, and $n$

  2️⃣ Substitute values into the formula $a_{n} =$$a_{1} *$$r^{(n - 1)}$

  3️⃣ Calculate the exponent $r^{(n - 1)}$

  4️⃣ Multiply $a_{1}$ by the result
• Match the type of sequence with its constant:

  Arithmetic Sequence ↔️ Difference

  Geometric Sequence ↔️ Ratio
• What is the key difference between arithmetic and geometric sequences?

  Terms change differently
• What type of sequence uses a common difference to generate terms?

  Arithmetic sequence
• What is the common ratio in the geometric sequence 3, 6, 12, 24, ...?

  2
• What is the definition of a series in mathematics?

  Sum of sequence terms
• Match the series type with its sum formula:

  Arithmetic Series ↔️ $S_{n} =$$\frac{n}{2}(2a_{1} +$$(n - 1)d)$

  Geometric Series ↔️ $S_{n} =$$\frac{a_{1}(1 - r^{n})}{1 - r}$
• What is a partial sum of an arithmetic series?

  Sum of specific terms
• What does $a_{n}$ represent in the arithmetic series partial sum formula?


  n-th term
• In the geometric series partial sum formula, 'n' represents the number of terms to sum.


  True
• What is an infinite geometric series?

  Sum of infinite terms
• The sum of the infinite geometric series 2, 1, 0.5, 0.25, ... is 4