Generalizing patterns with expressions and equations: แบบฝึกหัด

In the sequence: \( 4, 7, 10, 13, \dots \), what is the rule for finding the \( n \)-th term?

Solve for \( y \) in the equation: \( 4y - 6 = 18 \).

Which of the following expressions represents "5 more than 3 times a number \( n \)"?

A rectangular garden has a length of \( L \) meters. The width is 4 meters shorter than the length. If the perimeter of the garden is 32 meters, which equation can be used to find \( L \)?

If \( x + 7 = 15 \), what is the value of \( x \)?

If \(x + 5 = 12\), what is the value of \(x\)?

The sum of three consecutive integers is 45. If the smallest integer is represented by \(n\), set up an equation and solve for \(n\).

In a school library, the number of fiction books is 15 more than twice the number of non-fiction books. If there are 165 fiction books, find the number of non-fiction books by setting up an algebraic equation.

Part a: A box contains some candies. If there are \( y \) candies in each box and Sarah has 5 such boxes, write an algebraic expression for the total number of candies she has.
Part b: If Sarah gives 12 candies to her brother, write an algebraic expression for the number of candies she has left in terms of \( y \).
Part c: If \( y = 20 \), calculate the actual number of candies Sarah has left after giving some to her brother. <\/p>

A rectangle has a length that is 4 cm longer than its width. Let the width be \( w \) cm.
Part a: Write an algebraic expression for the length of the rectangle in terms of \( w \).
Part b: Write an algebraic expression for the perimeter of the rectangle in terms of \( w \) and simplify it.
Part c: If the width is 6 cm, find the actual perimeter of the rectangle.
Part d: If the perimeter is 40 cm, write an equation to represent this and find the value of \( w \).<\/p>