Question

The number of squares grows with each step in the pattern as shown below. How many squares are in Step n? Hint: use the table & your calculator 고 Step 1 Step 2 Step 3 Step 1 # of squares 2 2 6 3 12 4 20 n

Answer 1

We have the following sequence:

step 1 has 2 squares

step 2 has 6 squares

step 3 has 12 squares

step 4 has 20 squares

We have to find the number of squares for step n.

The oprtions are:

x^2

x^2+x

2x

x^2+1

If for step 1, we have 2 squares, the option x^2 is discarded as it would predict 1 square.

For step 2 we have 6 squares, so we can test the options:

`\begin{gathered} x=2 \\ \Rightarrow x^2+x=2^2+2=4+2=6\longrightarrow\text{ Possible solution} \\ \Rightarrow2x=2\cdot2=4\\eq6\longrightarrow\text{Discarded} \\ \Rightarrow x^2+1=2^2+1=4+1=5\\eq6\longrightarrow\text{Discarded} \end{gathered}`

The only option that satisfies step 2 is x^2+x.

We will test it with step 3 and 4:

`\begin{gathered} \text{Step 3} \\ x=3\longrightarrow x^2+x=3^2+3=9+3=12 \\ \text{Step 4} \\ x=4\longrightarrow x^2+x=4^2+4=16+4=20 \end{gathered}`

Both are correct, so we can conclude that for step n, the number of squares will be n^2+n

Answer: x^2+x