Isolate k from the given expression:
[tex]n=a+(k-1)d[/tex]
Substract a from both sides of the equation:
[tex]\begin{gathered} n-a=a+(k-1)d-a \\ \Rightarrow n-a=(k-1)d \end{gathered}[/tex]
Divide both members of the equation by d:
[tex]\begin{gathered} \Rightarrow\frac{n-a}{d}=\frac{(k-1)d}{d} \\ \Rightarrow\frac{n-a}{d}=k-1 \end{gathered}[/tex]
Add 1 to both sides of the equation:
[tex]\begin{gathered} \Rightarrow\frac{n-a}{d}+1=k-1+1 \\ \Rightarrow\frac{n-a}{d}+1=k \\ \end{gathered}[/tex]
Write 1 as d/d:
[tex]\begin{gathered} \Rightarrow k=\frac{n-a}{d}+1 \\ =\frac{n-a}{d}+\frac{d}{d} \\ =\frac{n-a+d}{d} \end{gathered}[/tex]
Therefore, the correct answer is displayed in option D:
[tex]k=\frac{n-a+d}{d}[/tex]
