Algebraic Expressions Patterns: Questions and Solutions - Grade 9

This page is designed to help students, parents, and teachers master the topic of algebraic expression patterns through carefully selected sequence questions and step-by-step solutions. Understanding how patterns function is a foundational skill in algebra, as it strengthens logical reasoning and prepares students for more complex functions.

The questions on this page will challenge you to:

Identify operational rules governing a sequence of algebraic terms (addition, subtraction, multiplication, and division).
Analyze shifting exponents and coefficients across sequential terms.
Extend algebraic patterns accurately to determine the next logical term in the sequence.

Questions

Identify the pattern and write the next term: \( 1, x^2 , x^4 , x^6 , x^8 , --- \)
Extend the pattern and write the next term: \( x^{10} , x^8 , x^6 , x^4 , x^2 , --- \)
Extend the pattern: \( x + y , 2 x + y , 3 x + y , 4 x + y , --- \)
Identify the pattern and write the next term: \(2x , 4x + 1, 6x + 2, 8x + 3, --- \)
Identify the pattern and find the next term: \( x + y + 1, 2 x + 3 y + 1 , 3 x + 5 y + 1 , 4 x + 7 y + 1 , --- \)
Identify the pattern and write the next term: \( x^3 + 10 x, x^3 + 9 x , x^3 + 8 x , x^3 + 7 x , --- \)
Extend the pattern and write the next term: \(3a + 2 , 6a + 5 , 12 a + 11 , 24 a + 23 , --- \)
Identify the pattern and write the next term: \( 1 , 2 x^2, 4 x^4, 8 x^6, 16 x^8 , --- \)
Extend the pattern and write the next term: \( x + y , x^2 y + y^2 x , x^3 y^2 + y^3 x^2 , x^4 y^3 + y^4 x^3 , --- \)
Extend the pattern and write the next term: \( x^{10} \; y^{10}, x^{11} \; y^{9} , x^{12} \; y^{8} , x^{13} \; y^{7}, --- \)

Step-by-Step Solutions with Explanations

Solution to Question 1
The pattern is obtained by multiplying terms by \( x^2 \) to obtain the next term.
- \( 1 \) : first term
- \( 1 \times \color{red}{x^2} = x^2 \)
- \( x^2 \times \color{red}{x^2} = x^{2+2} = x^4 \)
- \( x^4 \times \color{red}{x^2} = x^{4+2} = x^6 \)
- \( x^6 \times \color{red}{x^2} = x^{6+2} = x^8 \)
Next term is: \[ x^8 \times \color{red}{x^2} = x^{8+2} = x^{10} \]
Solution to Question 2
The pattern is obtained by dividing by \( x^2 \) to obtain the next term.
- \( x^{10} \) : first term
- \( \dfrac{x^{10}}{\color{red}{x^2}} = x^{10-2} = x^8 \)
- \( \dfrac{x^{8}}{\color{red}{x^2}} = x^{8-2} = x^6 \)
- \( \dfrac{x^{6}}{\color{red}{x^2}} = x^{6-2} = x^4 \)
- \( \dfrac{x^{4}}{\color{red}{x^2}} = x^{4-2} = x^2 \)
Next term is: \[ \dfrac{x^{2}}{\color{red}{x^2}} = 1 \]
Solution to Question 3
The pattern is obtained by adding \( x \) to obtain the next term.
- \( x + y \) : first term
- \( x + y + \color{red}{x} = 2 x + y \)
- \( 2 x + y + \color{red}{x} = 3 x + y \)
- \( 3 x + y + \color{red}{x} = 4 x + y \)
Next term is: \[ 4 x + y + \color{red}{x} = 5x + y \]
Solution to Question 4
The pattern is obtained by adding \( 2x + 1 \) to obtain the next term.
- \( 2 x \) : first term
- \( 2x + \color{red}{2 x + 1} = 4x + 1 \)
- \( 4x + 1 + \color{red}{2 x + 1} = 6x + 2 \)
- \( 6x + 2 + \color{red}{2 x + 1} = 8 x + 3 \)
Next term is: \[ 8 x + 3 + \color{red}{2 x + 1} = 10 x + 4 \]
Solution to Question 5
The pattern is obtained by adding \( x + 2y \) to obtain the next term.
- \( x + y + 1 \) : first term
- \( x + y + 1 + \color{red}{x + 2y} = 2x + 3y + 1 \)
- \( 2x + 3y + 1 + \color{red}{x + 2y} = 3x + 5y + 1 \)
- \( 3x + 5y + 1 + \color{red}{x + 2y} = 4x + 7y + 1 \)
Next term is: \[ 4x + 7y + 1 + \color{red}{x + 2y} = 5x + 9y + 1 \]
Solution to Question 6
The pattern is obtained by subtracting \( x \) to obtain the next term.
- \( x^3 + 10 x \) : first term
- \( x^3 + 10 x \color{red}{- x} = x^3 + 9x \)
- \( x^3 + 9x \color{red}{- x} = x^3 + 8 x \)
- \( x^3 + 8x \color{red}{- x} = x^3 + 7 x \)
Next term is: \[ x^3 + 7 x \color{red}{- x} = x^3 + 6x \]
Solution to Question 7
The pattern is obtained by doubling the expression and adding \( 1 \) to obtain the next term.
- \( 3a + 2 \) : first term
- \( \color{red}{2}(3a + 2) \color{red}{+ 1} = 6a + 4 + 1 = 6a + 5 \)
- \( \color{red}{2}(6a + 5) \color{red}{+ 1} = 12a + 10 + 1 = 12a + 11 \)
- \( \color{red}{2}(12a + 11) \color{red}{+ 1} = 24a + 22 + 1 = 24a + 23 \)
Next term is: \[ \color{red}{2}(24a + 23) \color{red}{+ 1} = 48a + 46 + 1 = 48a + 47 \]
Solution to Question 8
The pattern is obtained by multiplying by \( 2 x^2 \) to obtain the next term.
- \( 1 \) : first term
- \( 1 \color{red}{\times 2 x^2} = 2 x^2 \)
- \( 2 x^2 \color{red}{\times 2 x^2} = 4 x^{2+2} = 4 x^4 \)
- \( 4 x^4 \color{red}{\times 2 x^2} = 8 x^{4+2} = 8 x^6 \)
- \( 8 x^6 \color{red}{\times 2 x^2} = 16 x^{6+2} = 16 x^8 \)
Next term is: \[ 16 x^8 \color{red}{\times 2 x^2} = 32 x^{8+2} = 32 x^{10} \]
Solution to Question 9
The pattern is obtained by multiplying by \( xy \) to obtain the next term.
- \( x + y \) : first term
- \( (x + y) \color{red}{xy} = x^2y + y^2x \)
- \( (x^2 y + y^2 x) \color{red}{xy} = x^3 y^2 + y^3 x^2 \)
- \( (x^3 y^2 + y^3 x^2) \color{red}{xy} = x^4 y^3 + y^4 x^3 \)
Next term is: \[ (x^4 y^3 + y^4 x^3) \color{red}{xy} = x^5 y^4 + y^5 x^4 \]
Solution to Question 10
The pattern is obtained by multiplying by \( \dfrac{x}{y} \) to obtain the next term.
- \( x^{10} \; y^{10} \) : first term
- \( x^{10} \; y^{10} \color{red}{\times \dfrac{x}{y}} = x^{10+1} \; y^{10-1} = x^{11} \; y^{9} \)
- \( x^{11} \; y^{9} \color{red}{\times \dfrac{x}{y}} = x^{11+1} \; y^{9-1} = x^{12}\; y^{8} \)
- \( x^{12} \; y^{8} \color{red}{\times \dfrac{x}{y}} = x^{12+1} \; y^{8-1} = x^{13} \; y^{7} \)
Next term is: \[ x^{13} \; y^{7} \color{red}{\times \dfrac{x}{y}} = x^{13+1} \; y^{7-1} = x^{14} \; y^{6} \]

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