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by fskilnik@GMATH » Tue Mar 12, 2019 11:38 am

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GMATH practice exercise (Quant Class 17)

Claudio wants to be well prepared for a long marathon that will occur in the near future. His running coach has decided that the training sessions must be done once every two days, starting with a 15-kilometer distance route, and adding exactly 500 meters (=half a kilometer) to the route at every new training session. Claudio´s coach believes at least 1800 kilometers must be run, during the whole training period, before someone is considered ready for this challenge. According to these assumptions, what is the minimum number of training sessions that Claudio needs for this long marathon preparation?

(A) 59
(B) 60
(C) 61
(D) 62
(E) 63

Answer: [spoiler]_____(C)__[/spoiler]
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by fskilnik@GMATH » Tue Mar 12, 2019 1:50 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 17)

Claudio wants to be well prepared for a long marathon that will occur in the near future. His running coach has decided that the training sessions must be done once every two days, starting with a 15-kilometer distance route, and adding exactly 500 meters (=half a kilometer) to the route at every new training session. Claudio´s coach believes at least 1800 kilometers must be run, during the whole training period, before someone is considered ready for this challenge. According to these assumptions, what is the minimum number of training sessions that Claudio needs for this long marathon preparation?

(A) 59
(B) 60
(C) 61
(D) 62
(E) 63
$$?\,\,\,:\,\,\,\min \,N\,\,{\rm{for}}\,\,{\rm{sum}}\,\,\, \ge \,\,\,1800\,$$
$$\left. \matrix{
{a_1} = 15 \hfill \cr
{a_2} = 15 + 1 \cdot {1 \over 2} \hfill \cr
{a_3} = 15 + 2 \cdot {1 \over 2} \hfill \cr
\vdots \hfill \cr
{a_N} = 15 + \left( {N - 1} \right) \cdot {1 \over 2}\,\,\, \hfill \cr} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{Arith}}{\rm{. Seq}}{\rm{.}}} \,\,\,\,\,1800\,\, \le \,\,\,\,N \cdot {1 \over 2}\left[ {15 + 15 + {{\left( {N - 1} \right)} \over 2}} \right] = {{N\left( {N + 59} \right)} \over 4}$$
$${\rm{Trying}}\,\,\left( B \right)\,\,N = 60\,\,\,\,\, \Rightarrow \,\,\,\,\,{{N\left( {N + 59} \right)} \over 4} = {{{{60}^2} + 60\left( {60 - 1} \right)} \over 4} = {{2 \cdot 3600 - 60} \over 4} = 1800 - 15\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left( C \right)$$
$$\left( * \right)\,\,N = 61\,\,\,\,\, \Rightarrow \,\,\,\,{{N\left( {N + 59} \right)} \over 4}\,\, = \,\,\,\underbrace {1800 - 15}_{{\rm{sum}}\,\,{\rm{for}}\,\,N\,\, = \,\,60}\,\,\, + \,\,\underbrace {15 + 60 \cdot {1 \over 2}}_{{a_{61}}}\,\,\, = 1830\,\, > \,\,1800$$

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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