1.4 Sequences and Series

• Order the types of sequences based on their defining characteristic:

  1️⃣ Arithmetic: Constant difference

  2️⃣ Geometric: Constant ratio

  3️⃣ Fibonacci: Sum of preceding terms
• In the geometric sequence formula, $r$ represents the common ratio
• A series is denoted as a_1 + a_2 + a_3 + ...


  True
• How is a series mathematically denoted?

  $a_{1} +$$a_{2} +$$a_{3} +$$...$
• The formula for the $n$-th term of an arithmetic sequence is a_{n} = a_{1} + (n - 1)d</latex>
• What is a sequence in mathematics?

  An ordered list of terms
• A sequence focuses on the cumulative sum of terms.

  False
• A geometric sequence is characterized by a constant ratio
• What is the formula for the nth term of an arithmetic sequence?

  $a_{n} =$$a_{1} +$$(n - 1)d$
• The formula for the nth term of a geometric sequence is $a_{n} =$$a_{1} +$$(n - 1)d$.


  False
• A series focuses on the cumulative sum
• Match the concept with its description:

  Sequence ↔️ An ordered list of terms

  Series ↔️ The sum of the terms in a sequence
• A geometric sequence involves multiplication by a constant ratio.

  True
• Match the type of sequence with its formula:

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

  Geometric ↔️ $a_{n} =$$a_{1} \cdot r^{(n - 1)}$

  Fibonacci ↔️ $a_{n} =$$a_{n - 1} +$$a_{n - 2}$
• The formula for the $n$-th term of a geometric sequence is $a_{n} =$$a_{1} \cdot r^{(n - 1)}$, where $r$ is the common ratio
• Arrange the types of sequences based on their defining characteristics:

  1️⃣ Arithmetic: Constant difference

  2️⃣ Geometric: Constant ratio

  3️⃣ Fibonacci: Sum of preceding two terms
• What does $r$ represent in the geometric sequence formula?


  Common ratio
• Summation notation uses the Greek letter sigma to indicate the sum of terms in a series.


  True
• The linearity property of summation states that \sum_{i = 1}^{n} (a_{i} + b_{i}) = \sum_{i = 1}^{n} a_{i} + \sum_{i = 1}^{n} b_{i}</latex>
• What is another name for summation notation?

  Sigma notation
• What is the formula for the sum of constants in summation notation?

  \sum_{i = 1}^{n} c = n \cdot c</latex>
• What is the formula for the sum of the first n</latex> terms of a geometric series?

  $S_{n} =$$a_{1}\left(\frac{1 - r^{n}}{1 - r}\right)$
• What is the formula for the sum of a convergent infinite geometric series?

  S = \frac{a_{1}}{1 - r}</latex>
• A sequence is an ordered list of terms.
• A series is the sum of the terms in a sequence


  True
• In an arithmetic sequence, each term increases or decreases by a constant difference


  True
• The formula for the $n$-th term of a geometric sequence is a_{n} = a_{1} \cdot r^{(n - 1)}</latex>, where $r$ is the common ratio
• The common ratio in a geometric sequence can be any real number except 0


  True
• The sum of constants property states that \sum_{i = 1}^{n} c = n \cdot c</latex>, where $c$ is a constant
• The sum of the first $n$ terms of a geometric series is $S_{n} =$$a_{1}\left(\frac{1 - r^{n}}{1 - r}\right)$, where $a_{1}$ is the first term
• What does the variable $a_{1}$ represent in series formulas?


  First term
• The common ratio in a geometric series is denoted by $r$.


  True
• An infinite geometric series converges if |r| < 1</latex>, which means the common ratio's absolute value is less than 1
• Give an example of a convergent infinite geometric series.

  1 + 0.5 + 0.25 + ...
• A town with an initial population of 50,000 growing by 3% annually will have approximately 57,382 residents after 5 years.


  True
• A series is the sum of the terms in a sequence
• What is an arithmetic sequence characterized by?

  Constant difference between terms
• Each term in a Fibonacci sequence is the sum of the two preceding terms.

  True
• The nth term of a geometric sequence is given by a_n = a_1 \cdot r^{(n-1)}</latex>, where $r$ is the common ratio
• What does $a_{1}$ represent in the arithmetic sequence formula?


  First term