Quadratic Sequences

7.3.3 Quadratic Sequences

7.3.3 • Quadratic Sequences

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.

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

Compare your answer:

Step 4 is a 4-by-6 rectangle.

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.

Compare your answer:

Kiran is correct that the number of rows and the number of columns each grow by 1, but the total number of squares does not increase by the same amount from step to step. So, the total number of squares does not grow linearly.

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

Compare your answer:

Diego’s expression $n (n + 2)$ is correct because there are $n$ columns and $n + 2$ rows of small squares in Step $n$ . Jada’s expression is also correct because the $n (n + 2)$ array of small squares can be divided into an $n$ -by- $n$ square (having $n^{2}$ small squares) and an $n$ -by-2 rectangle (having $2 n$ small squares).

Three rectangles: the first is split horizontally into two sections labeled n across the top and bottom. Along the side of the first rectangle, it is labeled n and 2. The total area n(n+2) is written underneath. The second and third rectangles are separated. One has been labeled as a square with side length n and the third rectangle is labled with side lengths 2 across the top and n along the side. Underneath the rectangles, the areas of n squared plus 2 times n is listed.

Video: Quadratic Sequences

Watch the following video to learn more about quadratic sequences.

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Self Check

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

$y = 2n$
$y=n^2+1$
$y=n^2+n$
$y = n + 1$

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 $n$ and the number of squares $y$ .

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
$n$	$n + 1$	$n + 1$	$(n + 1)^{2} - 2$

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

$y = (n + 1)^{2} - 2$

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 $n$ and the number of squares $y$ .

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

Here is how to represent the quadratic sequence with an equation:

Notice there are no missing squares. Complete the table for each step:

Step	# of columns	# of rows	# of squares
1	4	4	16
2	5	5	25
3	6	6	36
$n$	$n + 3$	$n + 3$	$(n + 3)^{2}$

The number of columns is always $n + 3$ , and the number of rows is $n + 3$ .

The number of squares will be represented by $y = (n + 3)^{2}$ .