Subjects

Algebra 2

Geometric Series

957

•

May 28, 2023

•

7 pages

Sequences, Series, Exponential Functions, and Logarithmic Functions - Class 11 Formulas, Worksheets, and Examples

Name: Knowunity
Brand: Knowunity

Mira

@miraa._noui

A comprehensive guide to sequences and series class 11and... Show more

<p>The study guide for Class 11 Sequences and Series covers the definition and notation of sequences, as well as various types of sequences

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:

Sum of functions: Σ $f(x) + g(x)$ = Σf $x$ + Σg $x$
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"

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

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I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

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Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

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Elisha

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Paul T

iOS user

Subjects

Algebra 2

Geometric Series

Algebra 2

•

957

•

May 28, 2023

•

7 pages

Sequences, Series, Exponential Functions, and Logarithmic Functions - Class 11 Formulas, Worksheets, and Examples

Mira

@miraa._noui

A comprehensive guide to sequences and series class 11 and related mathematical concepts, focusing on fundamental principles, formulas, and practical applications.

• The guide covers essential topics including sequences and series formulas, exponential functions, logarithms, and their properties
•... Show more

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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:

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

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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

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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

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

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 $0,∞$ .

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Improve your grades

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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: $-∞,∞$

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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 $N$

Sign up to see the contentIt's free!

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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"

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user