Answer:
E) 2(n + 1)
Step-by-step explanation:
To find the expression for the number of straight lines needed to draw an n × n square grid, we first need to examine the number of straight lines needed for other square grids to see if a pattern emerges.
To draw a 1 × 1 square grid, we need 4 straight lines.
To draw a 3 × 3 square grid, we need 8 straight lines.
To draw a 2 × 2 square grid, we need 6 straight lines.
To draw a 4 × 4 square grid, we need 10 straight lines.
To draw a 5 × 5 square grid, we need 12 straight lines.
So, we can see that as n increases by 1, the number of straight lines needed increases by 2. As the difference is constant, the number of straight lines needed forms an arithmetic sequence.
The general form of the nth term of an arithmetic sequence is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of the $n$th term of an arithmetic sequence}}\\\\a_n=a+(n-1)d\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a_n$ is the nth term.}\\ \phantom{ww}\bullet\;\textsf{$a$ is the first term.}\\\phantom{ww}\bullet\;\textsf{$d$ is the common difference between terms.}\\\phantom{ww}\bullet\;\textsf{$n$ is the position of the term.}\\\end{array}}[/tex]
In this case, the first term (when n = 1) is a = 4, since it takes 4 straight lines to draw a 1 × 1 square grid. The common difference is d = 2, because as n increases by 1, the number of straight lines needed increases by 2. Therefore, the formula for the nth term of the sequence is:
[tex]a_n=4+(n-1)2[/tex]
Simplify:
[tex]a_n=4+2n-2[/tex]
[tex]a_n=2n+2[/tex]
Factor out the common term 2:
[tex]a_n=2(n+1)[/tex]
Therefore, the expression for the number of straight lines needed to draw an n × n square grid is:
[tex]\huge\boxed{\boxed{2(n+1)}}[/tex]
