Notes on Sequences, Exponential and Logarithmic Functions

Introduction to Sequences

• Learning Outcomes:

  • Define a sequence.

  • Recognize sequences in explicit or recursive forms.

  • Find terms in a sequence.

  • Expand a sum.

  • Use summation notation.

Types of Sequences

Explicit and Recursive Forms

• Explicit Form: Defines the nth term directly (e.g., $s_n = n$).

• Recursive Form: Defines the nth term based on previous terms (e.g., $s<em>n = s</em>{n-1} + d$).

Examples

1. **Explicit Sequences:

   • $s_n = n$ (first five terms: 1, 2, 3, 4, 5)

   • $s_n = 2n - 3$ (first five terms: -1, 1, 3, 5, 7)

   • $s_n = 1 - n^2$ (first five terms: 0, -3, -8, -15, -24)

   • $s_n = 2^{n+1}$ (first five terms: 4, 8, 16, 32, 64)

2. Recursive Sequences:

   • Given $a<em>1 = 2$, $a</em>n = 3 + a_{n-1}$ produces:

     • First five terms: 2, 5, 8, 11, 14.

Arithmetic Sequences

Recognition and Key Features

• An arithmetic sequence has a common difference (d) between consecutive terms.

• Formula of n-th term: $a<em>n = a</em>1 + (n - 1)d$

  • Where $a_1$ is the first term and $d$ is the common difference.

Examples:

1. Given sequence: $(4, 6, 8, 10)$.

   • Common difference: $d = 2$.

Finding Specific Terms

Methodology

1. Determining nth Term:

   • E.g., find the 41st term for $2, 6, 10, 14,…$

     • First term ($a_1$): 2, Common difference ($d$): 4.

     • Use: $a<em>n = a</em>1 + (n - 1)d <br>ightarrow a_{41} = 2 + (41 - 1)4 = 162$.

2. Finding common difference of a sequence:

   • E.g., For $s_n = (3n + 5)$, the common difference $d = 3$.

Summation of Sequences

Formulas for Sums

1. Sum of the First n Terms of an Arithmetic Sequence:

   • $S<em>n = rac{n}{2} (a</em>1 + a<em>n)$ or $S</em>n = rac{n}{2} [2a_1 + (n - 1)d]$

2. Examples of Summation:

   • E.g., Find $S_n$ for $60 + 64 + 68 + … + 120$:

     • Solve for $n$, then apply summation formulas.

Geometric Sequences

Recognition and Properties

• A geometric sequence has a common ratio (r).

• nth Term Formula:
  $a<em>n = a</em>1 r^{n-1}$.

Examples:

1. Given sequence: $1, 2, 4, 8$

   • Common ratio $r = 2$.

Exponential Functions

Key Features and Graphing

• Exponential Growth: $f(x) = a imes b^x$, b > 1.

• Exponential Decay: $f(x) = a imes b^x$, 0 < b < 1.

• Sample exponential function: $f(x) = 2^x$, graphing shows key features (domain, range, and asymptotes).

Logarithmic Functions

Definitions and Properties

• Logarithmic functions are inverses of exponential functions.

• Key Properties:

  • $ext{Domain} = (0, ext{∞})$, $ext{Range} = (- ext{∞}, ext{∞})$.

  • Vertical asymptote at $x = 0$.

Converting Logarithmic to Exponential Forms

• If $f(x) = ext{log}_b a$, then it can be written as $b^f(x) = a$.

Applications

• Both exponential and logarithmic functions allow for practical application in growth models, decay analysis, and financial calculations.

• Examples revolve around real-world applications in finance (interest rates), biology (population growth), and physics (radioactive decay).