Problem Solving

BTGmoderatorDC wrote:In the sequence S, each term after the first is twice the previous term. If the first term of sequence S is 3, what is the sum of the 14th, 15th, and 16th terms in sequence S?

(A) 3(2^16)
(B) 9(2^15)
(C) 21(2^14)
(D) 9(2^14)
(E) 21(2^13)
Source: Manhattan Prep

\[S \to \,\,\,{a_{\,1}}\,,\,\,{a_{\,2}}\,,\,\,{a_{\,3}}\,,\,\,\, \ldots \,\,\,\,\left\{ \begin{gathered}
\,{a_{\,1}} = 3 \hfill \\
{a_{\,n}} = 2 \cdot {a_{\,n - 1}}\,\,\,\,\left( {n \geqslant 2} \right) \hfill \\
\end{gathered} \right.\]
\[? = {a_{\,14}} + {a_{\,15}} + {a_{\,16}}\]

The PATTERN is clear:
\[{a_{\,\boxed2}} = {2^{\,\boxed1}} \cdot {a_{\,1}}\,\,\,\,\,\,;\,\,\,\,\,\,{a_{\,\boxed3}} = 2 \cdot {a_{\,2}} = {2^{\,\boxed2}} \cdot {a_{\,1}}\,\,\,\,\,\,;\,\,\,\,\,\,{a_{\,\boxed4}} = 2 \cdot {a_{\,3}} = {2^{\,\boxed3}} \cdot {a_{\,1}}\,\,\,\,\,\,;\,\,\,\,\,\, \ldots \]
\[\left. \begin{gathered}
{a_{\,14}} = {2^{\,13}} \cdot {a_{\,1}} \hfill \\
{a_{\,15}} = 2 \cdot {2^{\,13}} \cdot {a_{\,1}}\,\, \hfill \\
{a_{\,16}} = {2^2} \cdot {2^{\,13}} \cdot {a_{\,1}} \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\, = \,\,\,{2^{13}} \cdot {a_1} \cdot \left( {1 + 2 + {2^{\,2}}} \right)\,\,\,\mathop = \limits^{{a_{\,1}}\, = \,\,3} \,\,\,\,3 \cdot 7 \cdot {2^{\,13}}\]


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

Regards,
fskilnik.