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Algebra 1

7.3.3 Quadratic Sequences

Algebra 17.3.3 Quadratic Sequences

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Activity

A pattern of squares. Step 1 has a column of three squares. Step 2 has two columns with four squares in each column. Step 3 has three columns with five squares in each column.
1.

Sketch the next step in the pattern. Explain what you sketched.

2.

Kiran says that the pattern is growing linearly because as the step number goes up by 1, the number of rows and the number of columns also increase by 1. Do you agree? Be prepared to show your reasoning.

3.

To represent the number of squares after nn steps, Diego and Jada wrote different equations. Diego wrote the equation f(n)=n(n+2)f(n)=n(n+2). Jada wrote the equation f(n)=n2+2nf(n)=n2+2n. Is either Diego or Jada correct? Be prepared to show your reasoning.

Video: Quadratic Sequences

Watch the following video to learn more about quadratic sequences.

Self Check

Which equation represents the relationship between the step number n and the number of squares y ?

  1. y = 2 n
  2. y = n 2 + 1
  3. y = n 2 + n
  4. y = n + 1

Additional Resources

More Complex Quadratic Patterns

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number nn and the number of squares yy.

A pattern of squares. Step 1 has two columns of squares in two rows, but there are only two squares. In the first column, the square is located in the top row. In the second column, the square is located in the bottom row. It makes the squares look like they are in a diagonal pattern with the bottom left square and the top right square missing. Step 2 has 3 columns of squares. In the first column, there are 2 squares. In the second column, there are 3 squares. The squares are aligned to the top of the squares in the previous column so it looks like a square is missing from the first column. Then, in the last column, there are only 2 squares. The squares in this last column align to the bottom of the squares in the previous column so it looks like there is a square missing from the top of the column. Step 3 has 4 columns of squares. The first column contains 3 squares and the next two columns contain 4 squares. The columns are arranged such that they all start at the same level so it looks like a square is missing from the first column. Then, in the last column there are 3 squares. The squares in this column align with the bottom of the previous two columns so it looks like there is a square missing from the top of the column.

First, try to find the pattern if the missing squares were actually there.

Make a table:

Step # of columns # of rows # of squares
1 2 2 2
2 2 2 7
3 4 4 14
nn n+1n+1 n+1n+1 (n+1)22(n+1)22

There would be (n+1)(n+1)(n+1)(n+1) squares, but there are 2 missing from each step, so the equation would be:

y=(n+1)22y=(n+1)22

Try it

Try It: More Complex Quadratic Patterns

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number nn and the number of squares yy.

A pattern of squares. Step 1 has  4 rows with 4 squares in each row. Step 2 has 5 rows with 5 squares in each row. Step 3 has 6 rows with 6 squares in each row

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