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

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GMATH Teacher
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If F is a function defined in the positive integers, such that F(k) is  [#permalink]

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10 Mar 2019, 08:09
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55% (hard)

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59% (01:42) correct 41% (01:27) wrong based on 39 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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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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If F is a function defined in the positive integers, such that F(k) is  [#permalink]

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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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MEMENTO AUDERE SEMPER
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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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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.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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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