2.1 Relating geometric sequences and exponential functions

• The formula for the nth term of a geometric sequence is a_{n}
• In the geometric sequence 2, 4, 8, 16, what is the common ratio?

  2
• To find the common ratio, divide the second term by the first
• What is the general formula for finding the nth term of a geometric sequence?

  a_{n} = a_{1} \cdot r^{n - 1}</latex>
• The general formula for the sequence 100, 50, 25, 12.5, ... is $a_{n} =$$100 \cdot (0.5)^{n - 1}$, where $r =$ 0.5
• In an exponential function, what does the base represent?

  Growth factor
• What is the nth term of the geometric sequence 2, 6, 18, 54, ... if $r =$$3$?


  $2 \cdot 3^{n - 1}$
• To find the common ratio in a geometric sequence, divide any term by the term immediately preceding
• The formula for the common ratio in a geometric sequence is $r =$$\frac{a_{n}}{a_{n - 1}}$, where $a_{n}$ is the nth term and $a_{n - 1}$ is the term immediately preceding
• The general formula for the nth term of a geometric sequence is $a_{n} =$$a_{1} \cdot r^{n - 1}$, where $a_{1}$ is the first term, $r$ is the common ratio, and $n$ is the term number
• What is the general form of an exponential function?

  $f(x) =$$ab^{x}$
• Match the feature with its corresponding type:

  Continuous ↔️ Exponential Function

  Discrete ↔️ Geometric Sequence
• The exponent in an exponential function indicates the power to which the base is raised.

  True
• Geometric sequences have a growth/decay factor called the common ratio $r$, while exponential functions have a growth/decay factor called the base $b$.


  True
• Geometric sequences and exponential functions can both describe exponential growth or decay.


  True
• The nth term of a geometric sequence can be found using the formula $a_{n} =$$a_{1} \cdot r^{n - 1}$.


  True
• An exponential function like f(x) = 2(3)^{x}</latex> grows continuously.


  True
• The 5th term in the sequence 2, 6, 18, 54, ... is 162.
• What is the term for the constant multiple between consecutive terms in a geometric sequence?

  Common ratio
• The common ratio in a geometric sequence is denoted by the letter r
• The general formula for the nth term of a geometric sequence is a_{n}
• Steps to find the common ratio in a geometric sequence

  1️⃣ Select two consecutive terms

  2️⃣ Divide the second term by the first

  3️⃣ Verify the ratio remains consistent
• In the general formula for a geometric sequence, $r$ represents the common ratio.


  True
• What is the general form of an exponential function?

  $f(x) =$$ab^{x}$
• Exponential functions are continuous, while geometric sequences are discrete
• What is the common ratio in the geometric sequence 4, 12, 36, 108, ...?

  3
• The common ratio in a geometric sequence must be the same across all pairs of consecutive terms.


  True
• Verifying the common ratio across multiple pairs of terms ensures its consistency in a geometric sequence.


  True
• In the formula $a_{n} =$$a_{1} \cdot r^{n - 1}$, $a_{n}$ represents the nth term of the geometric sequence.


  True
• In the exponential function $f(x) =$$ab^{x}$, $a$ represents the initial value
• What is the role of the base in an exponential function?

  Growth or decay rate
• What does the variable $a$ represent in the general formula for exponential functions?


  Initial value
• What is the primary difference between geometric sequences and exponential functions in terms of continuity?

  Discrete vs. Continuous
• What is the common ratio in a geometric sequence defined as?

  Constant multiple
• A geometric sequence is a type of discrete function.
• In a geometric sequence, each term is a constant multiple of the previous term called the common ratio.
• Geometric sequences are discrete, while exponential functions are continuous.


  True
• What is the constant called in a geometric sequence that each term is multiplied by?

  Common ratio
• What is the general form of an exponential function?

  $f(x) =$$ab^{x}$
• What is the formula to calculate the common ratio in a geometric sequence?

  $r =$$\frac{a_{n}}{a_{n - 1}}$