Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
The terms of a sequence a1, a2, a3, ... are defined by an=an-2+3. Is 411 a term of the sequence?
1) a1=111
2) a2=112
Very nice problem, Max. Congrats!
$$S\,\,{\rm{sequence:}}\,\,\left\{ \matrix{
{{\rm{a}}_{\rm{1}}},{a_2}, \ldots \hfill \cr
{a_n} = {a_{n - 2}} + 3\,\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,n \ge 3 \hfill \cr} \right.\,\,\,\,\,\left( * \right)$$
$$411\,\,\mathop \in \limits^? \,\,\,S$$
$$\left( 1 \right)\,\,{a_1} = 111\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,{a_3} = 111 + 3\,\,\,\, \Rightarrow \,\,\,{a_5} = 111 + 3 + 3 = 111 + 2 \cdot 3\,\,\, \Rightarrow \,\,\,\,{a_7} = 111 + 3 \cdot 3\,\,\,\, \Rightarrow \,\,\, \ldots $$
$${\text{Hence}}\,\,411 = 111 + 100 \cdot 3\,\, \in \,\,\,S\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,{a_2} = 112\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,{a_4} = 112 + 3\,\,\,\, \Rightarrow \,\,\,{a_6} = 112 + 2 \cdot 3\,\,\, \Rightarrow \,\,\,\, \ldots $$
$$411 \ne 112 + k \cdot 3\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,k\,\, \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,{{\rm{a}}_{\rm{1}}}{\rm{ = 111}}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,Take\,\,{a_1} = 112\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
Obs.: note that defining a
1 and a
2 , the sequence S is uniquely defined.
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
