Questions on identifying patterns of a algebraic expressions are presented along with their solutions.
Questions
Question 1: Identify the pattern and write the next term: \( 1, x^2 , x^4 , x^6 , x^8 , --- \).
Question 2: Extend the pattern and write the next term: \( x^{10} , x^8 , x^6 , x^4 , x^2 , --- \).
Question 3: Extend the pattern: \( x + y , 2 x + y , 3 x + y , 4 x + y , --- \).
Question 4: Identify the pattern and write the next term: \(2x , 4x + 1, 6x + 2, 8x + 3, --- \).
Question 5: Identify the pattern and find the next term: \( x + y + 1, 2 x + 3 y + 1 , 3 x + 5 y + 1 , 4 x + 7 y + 1 , --- \).
Question 6: Identify the pattern and write the next term: \( x^3 + 10 x, x^3 + 9 x , x^3 + 8 x , x^3 + 7 x , --- \).
Question 7: Extend the pattern and write the next term: \(3a + 2 , 6a + 5 , 12 a + 11 , 24 a + 23 , --- \).
Question 8: Identify the pattern and write the next term:: \( 1 , 2 x^2, 4 x^4, 8 x^6, 16 x^8 , --- \).
Question 9: Extend the pattern and write the next term:: \( x + y , x^2 y + y^2 x , x^3 y^2 + y^3 x^2 , x^4 y^3 + y^4 x^3 , ---\).
Question 10: Extend the pattern and write the next term:: \( x^{10} \; y^{10}, x^{11} \; y^{9} , x^{12} \; y^{8} , x^{13} \; y^{7}, --- \).
Solutions
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 terms 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 terms is \[ 8 x + 3 + \color{red}{2 x + 1} = 10 x + 4\].
Solution to Question 5: The pattern is obtained by adding \( x + 2 y \) to obtain the next term.
\( x + y + 1 \) : first term
\( x + y + 1 + \color{red}{x + 2y} = 2x + 3y + 1 \)
\( 2x + 3y + 1 + \color{red}{x + 2y} = 3 x + 5y + 1\)
\( 3 x + 5y + 1 + \color{red}{x + 2y} = 4x + 7y + 1\)
Next terms is \[ 4x + 7y + 1 + \color{red}{x + 2y} = 5 x + 9y + 1\].
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 and adding \( 1 \) to obtain the next term.
\( 3a + 2 \) : first term
\( \color{red}{2}(3a + 2) \color{red}{+1} = 6 a + 4 + 1 = 6 a + 5 \)
\( \color{red}{2}(6 a + 5) \color{red}{+1} = 12 a + 10 + 1 = 12 a + 11 \)
\( \color{red}{2}(12 a + 11) \color{red}{+1} = 24 a + 22 + 1 = 24 a + 23 \)
Next term is \[ \color{red}{2}(24 a + 23) \color{red}{+1} = 48 a + 46 + 1 = 48 a + 47 \]
Solution to Question 8: The pattern is obtained by mutliplying 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 mutliplying by \( x y \) to obtain the next term.
\( x + y \) : first term
\( (x + y) \color{red}{ x y} = x^2 y + y^2 x \)
\( (x^2 y + y^2 x) \color{red}{ x y} = x^3 y^2 + y^3 x^2 \)
\( (x^3 y^2 + y^3 x^2) \color{red}{ x y} = x^4 y^3 + y^4 x^3 \)
Next terms is \[ (x^4 y^3 + y^4 x^3) \color{red}{ x y} = x^5 y^4 + y^5 x^4 \].
Solution to Question 10: The pattern is obtained by mutliplying 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 terms is \[ x^{13} \; y^{7} \color{red}{ \times \dfrac{x}{y} } = x^{13+1} \; y^{7-1} = x^{14} \; y^{6} \]
