seofah
Mary persuaded n friends to donate $500 each to her election campaign, and then each of these n friends persuaded n more people to donate $500 each to Mary’s campaign. If no one donated more than once and if there were no other donations, what was the value of n?
(1) The first n people donated 1/16 of the total amount donated.
(2) The total amount donated was $120,000.
\({\rm{Total}}\,\, = \,\,500 \cdot n + 500 \cdot n \cdot n\,\,\,\,\,\,\left[ \$ \right]\)
\(? = n\)
\(\left( 1 \right)\,\,\,500 \cdot n = {1 \over {16}} \cdot 500 \cdot n \cdot \left( {1 + n} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\,\left( {500\,n} \right)\,\,\,\left[ {\,n\, \ne \,0\,} \right]} \,\,\,1 = {1 \over {16}} \cdot \left( {1 + n} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)
\(\left( 2 \right)\,\,\,500 \cdot n\left( {1 + n} \right) = 120000\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,500} \,\,\,\,n\left( {1 + n} \right) = 240\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,n\,\, > 0\,\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)
\(\left( * \right)\,\,15 \cdot 16 = 240\,\,\, \Rightarrow \,\,\,\left\{ \matrix{\\
\,n\left( {n + 1} \right) < 240\,\,{\rm{for}}\,\,0 < n < 15 \hfill \cr \\
\,n\left( {n + 1} \right) > 240\,\,{\rm{for}}\,\,n \ge 16 \hfill \cr} \right.\,\,\,\,\,\,\left( {{\rm{Now}}\,\,{\rm{rethink}}\,\,{\rm{without}}\,\,{\rm{knowing}}\,\,{\rm{that}}\,\,n = 15...} \right)\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.