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If F is a function defined in the positive integers, such that F(k) is

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New post 10 Mar 2019, 08:09
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55% (01:41) correct 45% (01:29) wrong based on 29 sessions

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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)

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New post 10 Mar 2019, 11:51
Statement 1: \(F(7+1) = 8 * F(7)\)

Going backwards, such as \(F (6 +1) = 7 * F (6)\) we notice that we aren't able to define F(1). \(F(1) = 1 * F(0)\), whereas \(0\) is a neutral number and \(F\) is not defined. Discard this option.

Statement 2: we have \(F(1) = 1\). Alone is insufficient, because it doesn't say anything else about how to compute F(8).

Since from statement 1 we lacked of \(F(1)\) value, combined together these two statements lead to solution. Hence, C.
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Re: If F is a function defined in the positive integers, such that F(k) is  [#permalink]

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New post 10 Mar 2019, 16:20
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fskilnik 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.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: If F is a function defined in the positive integers, such that F(k) is   [#permalink] 10 Mar 2019, 16:20
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