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$17$ . Nebula used black and white square tiles to form patterns as shown below. She recorded the number of black and white tiles used in the table. $\newline$ Pattern $1$ $\newline$ Pattern $2$ $\newline$ Pattern $3$ $\newline$ \begin{tabular}{|c|c|c|c|} $\newline$ \hline & \begin{tabular}{c} $\newline$ Total Number of \\ $\newline$ tiles $\newline$ \end{tabular} & \begin{tabular}{c} $\newline$ Number of Black \\ $\newline$ tiles $\newline$ \end{tabular} & \begin{tabular}{c} $\newline$ Number of White \\ $\newline$ tiles $\newline$ \end{tabular} \\ $\newline$ \hline Pattern $1$ & $3$ & $2$ & $1$ \\ $\newline$ \hline Pattern $2$ & $6$ & $3$ & $3$ \\ $\newline$ \hline Pattern $3$ & $9$ & $5$ & $4$ \\ $\newline$ \hline Pattern $4$ & $12$ & $6$ & $6$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ (a) Complete the table above for Pattern $4$ . [ $1$ ] $\newline$ (b) How many black tiles did Nebula use in Pattern $85$ ?

Full solution

17

. Nebula used black and white square tiles to form patterns as shown below. She recorded the number of black and white tiles used in the table.

\newline

Pattern

1

\newline

Pattern

2

\newline

Pattern

3

\newline

\begin{tabular}{|c|c|c|c|}

\newline

\hline & \begin{tabular}{c}

\newline

Total Number of \\

\newline

tiles

\newline

\end{tabular} & \begin{tabular}{c}

\newline

Number of Black \\

\newline

tiles

\newline

\end{tabular} & \begin{tabular}{c}

\newline

Number of White \\

\newline

tiles

\newline

\end{tabular} \\

\newline

\hline Pattern

1

3

2

1

\newline

\hline Pattern

2

6

3

3

\newline

\hline Pattern

3

9

5

4

\newline

\hline Pattern

4

12

6

6

\newline

\hline

\newline

\end{tabular}

\newline

(a) Complete the table above for Pattern

4

. [

1

]

\newline

(b) How many black tiles did Nebula use in Pattern

85

Pattern $4$ Analysis: For Pattern $4$ , we notice the number of black tiles increases by $1$ and the number of white tiles increases by $2$ from the previous pattern.
Calculating Black Tiles: Pattern $3$ has $5$ black tiles, so Pattern $4$ will have $5 + 1 = 6$ black tiles.
Calculating White Tiles: Pattern $3$ has $4$ white tiles, so Pattern $4$ will have $4 + 2 = 6$ white tiles.
Total Tiles in Pattern $4$ : The total number of tiles for Pattern $4$ is the sum of black and white tiles, which is $6 + 6 = 12$ .
Finding Black Tiles in Pattern $85$ : For Pattern $85$ , we need to find the number of black tiles. The pattern shows that the number of black tiles starts at $2$ and increases by $1$ for each subsequent pattern.
Finding Black Tiles in Pattern $85$ : For Pattern $85$ , we need to find the number of black tiles. The pattern shows that the number of black tiles starts at $2$ and increases by $1$ for each subsequent pattern.To find the number of black tiles in Pattern $85$ , we start with the $2$ black tiles in Pattern $1$ and add $1$ tile for each pattern up to Pattern $85$ : $2 + (85 - 1) = 2 + 84 = 86$ black tiles.