If n belongs to the set A, then n+5 belongs to the set A. Does 50 belong to the set A?

1) 30 belongs to the set A.
2) 100 belongs to the set A.

Another option: If n is a member of the set A, then n+5 is a member of the set A. Is 50 a member of the set A?
Etc.

\[n \in A\,\,\,\, \Rightarrow \,\,\,\,n + 5\,\, \in \,\,A\,\,\,\,\left( * \right)\]
\[50\,\,\,\mathop \in \limits^? \,\,\,A\]
\[\left( 1 \right)\,\,\,30 \in A\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,35 \in A\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,40 \in A\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,45 \in A\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,100 \in A\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,105 \in A\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,110 \in A\,\,\,\, \ldots \,\,\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,A = \left\{ {45,50,55,60, \ldots } \right\}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,A = \left\{ {100,105,110, \ldots } \right\}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\,\,\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Condition 1)
Since 30 is in the set A, 35, 40, 45, 50 are in the set A.
Condition 1) is sufficient.

Condition 2)
A = { 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, ... }
The set A has 40.
But the set A = { 100, 105, 110, ... } does not have 40.
Condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A