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Questions and Answers
What is the definition of a variable in algebra?
What is the definition of a variable in algebra?
What is the highest power of the variable in a linear equation?
What is the highest power of the variable in a linear equation?
How can quadratic equations be solved?
How can quadratic equations be solved?
What is the formula to find the nth term of an arithmetic progression?
What is the formula to find the nth term of an arithmetic progression?
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What is the sum of the first n terms of an arithmetic progression?
What is the sum of the first n terms of an arithmetic progression?
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What is the 10th term of the arithmetic progression 3, 6, 9, 12,...?
What is the 10th term of the arithmetic progression 3, 6, 9, 12,...?
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What is the common property of all arithmetic progressions?
What is the common property of all arithmetic progressions?
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What type of equation is x^3 - 2x^2 + x + 1 = 0?
What type of equation is x^3 - 2x^2 + x + 1 = 0?
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What is the degree of the polynomial equation x^3 - 2x^2 + x + 1 = 0?
What is the degree of the polynomial equation x^3 - 2x^2 + x + 1 = 0?
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What is the characteristic of an arithmetic progression that distinguishes it from other types of sequences?
What is the characteristic of an arithmetic progression that distinguishes it from other types of sequences?
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Study Notes
Algebra
Definitions
- Algebra: a branch of mathematics that deals with variables and their relationships
- Variable: a symbol that represents a value that can change
- Constant: a symbol that represents a fixed value
- Expression: a combination of variables, constants, and mathematical operations
- Equation: a statement that says two expressions are equal
Types of Equations
- Linear equation: an equation in which the highest power of the variable is 1
- Example: 2x + 3 = 5
- Quadratic equation: an equation in which the highest power of the variable is 2
- Example: x^2 + 4x + 4 = 0
- Polynomial equation: an equation consisting of variables and coefficients combined using only addition, subtraction, and multiplication
- Example: x^3 - 2x^2 + x + 1 = 0
Solving Equations
- Linear equations: can be solved using addition, subtraction, multiplication, and division
- Quadratic equations: can be solved using factoring, quadratic formula, or completing the square
- Polynomial equations: can be solved using factoring, remainder theorem, or synthetic division
Arithmetic Progression
Definition
- Arithmetic progression (AP): a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term
- Common difference (d): the fixed constant added to each term to get the next term
Formulae
- nth term of an AP: an = a1 + (n - 1)d
- Sum of first n terms of an AP: Sn = n/2 [2a1 + (n - 1)d]
Properties
- The common difference is constant
- The sequence is infinite
- The terms can be positive, negative, or zero
Examples
- 2, 5, 8, 11, ... is an AP with a1 = 2 and d = 3
- Find the 10th term of the AP: 3, 6, 9, 12, ...
Algebra
Types of Equations
- A linear equation has the highest power of the variable as 1, e.g., 2x + 3 = 5
- A quadratic equation has the highest power of the variable as 2, e.g., x^2 + 4x + 4 = 0
- A polynomial equation consists of variables and coefficients combined using addition, subtraction, and multiplication, e.g., x^3 - 2x^2 + x + 1 = 0
Solving Equations
- Linear equations can be solved using addition, subtraction, multiplication, and division
- Quadratic equations can be solved using factoring, quadratic formula, or completing the square
- Polynomial equations can be solved using factoring, remainder theorem, or synthetic division
Arithmetic Progression
Key Concepts
- An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term
- The fixed constant is called the common difference (d)
- The sequence is infinite, and terms can be positive, negative, or zero
Formulae
- The nth term of an AP is an = a1 + (n - 1)d
- The sum of first n terms of an AP is Sn = n/2 [2a1 + (n - 1)d]
Examples and Applications
- The sequence 2, 5, 8, 11,...is an AP with a1 = 2 and d = 3
- To find the 10th term of the AP: 3, 6, 9, 12,..., use the formula an = a1 + (n - 1)d
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Description
Learn the basics of algebra including definitions of variables, constants, expressions, and equations, as well as different types of equations like linear equations.
