Questions and Answers
What is the definition of a variable in algebra?
What is the highest power of the variable in a linear equation?
How can quadratic equations be solved?
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?
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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?
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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?
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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.