Claudio wants to be well prepared for a long marathon that will occur

\(?\,\,\,:\,\,\,\min \,N\,\,{\rm{for}}\,\,{\rm{sum}}\,\,\, \ge \,\,\,1800\,\)

\(\left. \matrix{\\
{a_1} = 15 \hfill \cr \\
{a_2} = 15 + 1 \cdot {1 \over 2} \hfill \cr \\
{a_3} = 15 + 2 \cdot {1 \over 2} \hfill \cr \\
\vdots \hfill \cr \\
{a_N} = 15 + \left( {N - 1} \right) \cdot {1 \over 2}\,\,\, \hfill \cr} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{Arith}}{\rm{. Seq}}{\rm{.}}} \,\,\,\,\,1800\,\, \le \,\,\,\,N \cdot {1 \over 2}\left[ {15 + 15 + {{\left( {N - 1} \right)} \over 2}} \right] = {{N\left( {N + 59} \right)} \over 4}\)

\({\rm{Trying}}\,\,\left( B \right)\,\,N = 60\,\,\,\,\, \Rightarrow \,\,\,\,\,{{N\left( {N + 59} \right)} \over 4} = {{{{60}^2} + 60\left( {60 - 1} \right)} \over 4} = {{2 \cdot 3600 - 60} \over 4} = 1800 - 15\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left( C \right)\)

\(\left( * \right)\,\,N = 61\,\,\,\,\, \Rightarrow \,\,\,\,{{N\left( {N + 59} \right)} \over 4}\,\, = \,\,\,\underbrace {1800 - 15}_{{\rm{sum}}\,\,{\rm{for}}\,\,N\,\, = \,\,60}\,\,\, + \,\,\underbrace {15 + 60 \cdot {1 \over 2}}_{{a_{61}}}\,\,\, = 1830\,\, > \,\,1800\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.