Page 2: Series and Summation Notation

This page delves into series calculations and sigma notation, providing fundamental properties and special series formulas.

Definition: A series is the sum of terms in a sequence, represented using sigma notation (Σ).

Vocabulary: Sigma notation includes:

• Index of summation $x$
• Lower bound (starting term)
• Upper bound (last term)

Example: The sum of constants: Σc from x=1 to n equals c·n

Highlight: Key summation properties include:

1. Sum of functions: Σ$f(x) + g(x)$ = Σf$x$ + Σg$x$
2. Constant multiplication: Σcf$x$ = cΣf$x$

Page 3: Advanced Series and Exponential Functions

This section covers arithmetic and geometric series, introducing exponential functions and their properties.

Definition: Exponential functions are those where the dependent variable increases/decreases by a constant multiple.

Example: Basic exponential function: y = a$b$ˣ

Highlight: For exponential functions:

• Domain is (-∞,∞)
• Range depends on 'a': (k,∞) if a>0; $-∞,k$ if a<0

Vocabulary: In exponential functions:

• b is the base/common ratio
• a is the initial/critical value
• h is horizontal shift
• k is vertical shift

Page 4: The Number e and Logarithms

This final section introduces the number e and logarithmic functions, connecting them to real-world applications.

Definition: e is an irrational number (≈2.718) serving as the natural base for logarithms.

Example: Compound interest formula: A = P$1 + r/n$ⁿ

Highlight: Logarithmic functions are inverses of exponential functions, used when solving for variables in exponents.

Quote: "Logarithmic functions are the inverses of exponential functions"

Vocabulary: In compound interest:

• A is account balance/future value
• P is principal/initial deposit
• r is annual interest rate
• t is time in years
• n is compounding frequency

Page 4: Exponential Functions and e

This section explores exponential functions and the number e pdf content, including practical applications.

Definition: The number e is an irrational number approximately equal to 2.718, often called Euler's number.

Example: Compound interest formula: A = P$1 + r/n$^(nt)

Highlight: The function y = e^x represents exponential growth with domain (-∞,∞) and range (0,∞).

Page 5: Logarithmic Functions

This page details logarithmic functions and properties worksheet concepts, focusing on graphing and transformations.

Definition: The general form of a logarithmic function is f$x$ = alog_b$x-h$ + k

Highlight: Key characteristics include:

• Vertical asymptote at x = h
• Domain: (h,∞)
• Range: (-∞,∞)

Page 6: Properties of Logarithms

This section covers essential properties of logarithms examples and fundamental concepts.

Definition: Key logarithmic properties include base cancellation and product rules.

Example: log_2$2^3$ = 3 (when bases are equal, the logarithm equals the exponent)

Highlight: The product rule states that log_a(MN) = log_a(M) + log_a(N)

Page 1: Fundamentals of Sequences

This page introduces the core concepts of sequences and their mathematical representation. The content focuses on both arithmetic and geometric sequences, providing essential formulas and notations.

Definition: A sequence is an ordered list of objects or numbers, with each item denoted as a "term".

Vocabulary: Domain of a sequence refers to consecutive list of terms (1, 2, 3, 4...n), while range represents the actual values in the sequence.

Example: For an arithmetic sequence 1, 3, 5, 7, the range is {1, 3, 5, 7}.

Highlight: Two key types of sequences are introduced:

• Arithmetic sequences with constant difference $d$
• Geometric sequences with constant ratio $r$

Quote: "The rule for arithmetic sequence: an = a₁ + d$n-1$ OR an = an-1 + d"