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If F is a function defined in the positive integers, such th

by fskilnik@GMATH » Sun Mar 10, 2019 6:53 am

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GMATH practice exercise (Quant Class 14)

If F is a function defined in the positive integers, such that F(k) is a positive integer for each positive integer k, what is the value of F(8)?

(1) F(n+1) = (n+1)*F(n), for every positive integer n.
(2) F(1)*F(1) = F(1)

Answer: [spoiler]_____(C)__[/spoiler]
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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If F is a function defined in the positive integers, such th

by fskilnik@GMATH » Sun Mar 10, 2019 3:11 pm

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 14)

If F is a function defined in the positive integers, such that F(k) is a positive integer for each positive integer k, what is the value of F(8)?

(1) F(n+1) = (n+1)*F(n), for every positive integer n.
(2) F(1)*F(1) = F(1)
$$F\left( k \right) \ge 1\,\,{\mathop{\rm int}} \,\,\,\,{\rm{for}}\,\,{\rm{each}}\,\,\,k \ge 1\,\,\,\,\left( * \right)$$
$$? = F\left( 8 \right)$$
$$\left( 1 \right)\,\,F\left( {n + 1} \right) = \left( {n + 1} \right) \cdot F\left( n \right)\,\,\,{\rm{for}}\,{\rm{each}}\,\,\,n \ge 1\,\,{\mathop{\rm int}} $$
$$\left. {\matrix{
{F\left( 2 \right) = 2 \cdot F\left( 1 \right)\,\,} \hfill \cr
{F\left( 3 \right) = 3 \cdot F\left( 2 \right) = 3 \cdot 2 \cdot F\left( 1 \right)} \hfill \cr
{\,\,\, \vdots } \hfill \cr
{? = F\left( 8 \right) = 8 \cdot 7 \cdot 6 \cdot \ldots \cdot 3 \cdot 2 \cdot F\left( 1 \right)\,\,\,} \hfill \cr

} } \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\{ {\matrix{
{\,{\rm{Take}}\,\,F\left( 1 \right) = 1\,\,\,\, \Rightarrow \,\,\,? = 8!\,\,} \hfill \cr
{\,{\rm{Take}}\,\,F\left( 1 \right) = 2\,\,\,\, \Rightarrow \,\,\,? = 2 \cdot 8!} \hfill \cr

} } \right.$$
$$\left( 2 \right)\,\,F\left( 1 \right) \cdot F\left( 1 \right) = F\left( 1 \right)\,\,\,\,\mathop \Rightarrow \limits^{\,:\,\,F\left( 1 \right)\,\, \ne \,0\,\,\left( * \right)} \,\,\,F\left( 1 \right) = 1$$
$$\left\{ {\matrix{
{\,{\rm{Take}}\,\,F\left( n \right) = 1\,\,\,{\rm{for}}\,{\rm{each}}\,\,\,n \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,? = 1\,\,} \hfill \cr
{\,{\rm{Take}}\,\,F\left( n \right) = \left( {n + 1} \right) \cdot F\left( n \right)\,\,\,{\rm{for}}\,{\rm{each}}\,\,\,n \ge 1\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,? = 8!\,} \hfill \cr
} } \right.$$
$$\left( {1 + 2} \right)\,\,\,\,? = 8!\,\,\,\,\, \Rightarrow \,\,\,\,\left( {\rm{C}} \right)$$

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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