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

Question

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)

lucajava · Answer

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 .

fskilnik · Answer

F\left( k ight) \ge 1\,\,{\mathop{m int}} \,\,\,\,{m{for}}\,\,{m{each}}\,\,\,k \ge 1\,\,\,\,\left( * ight)

? = F\left( 8 ight)

\left( 1 ight)\,\,F\left( {n + 1} ight) = \left( {n + 1} ight) \cdot F\left( n ight)\,\,\,{m{for}}\,{m{each}}\,\,\,n \ge 1\,\,{\mathop{m int}}

\left. {\matrix{
 {F\left( 2 ight) = 2 \cdot F\left( 1 ight)\,\,} \hfill \cr 
 {F\left( 3 ight) = 3 \cdot F\left( 2 ight) = 3 \cdot 2 \cdot F\left( 1 ight)} \hfill \cr 
 {\,\,\, \vdots } \hfill \cr 
 {? = F\left( 8 ight) = 8 \cdot 7 \cdot 6 \cdot \ldots \cdot 3 \cdot 2 \cdot F\left( 1 ight)\,\,\,} \hfill \cr

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

} } ight.

\left( 2 ight)\,\,F\left( 1 ight) \cdot F\left( 1 ight) = F\left( 1 ight)\,\,\,\,\mathop \Rightarrow \limits^{\,:\,\,F\left( 1 ight)\,\, 
e \,0\,\,\left( * ight)} \,\,\,F\left( 1 ight) = 1

\left\{ {\matrix{
 {\,{m{Take}}\,\,F\left( n ight) = 1\,\,\,{m{for}}\,{m{each}}\,\,\,n \ge 1\,\,{\mathop{m int}} \,\,\,\,\, \Rightarrow \,\,\,? = 1\,\,} \hfill \cr 
 {\,{m{Take}}\,\,F\left( n ight) = \left( {n + 1} ight) \cdot F\left( n ight)\,\,\,{m{for}}\,{m{each}}\,\,\,n \ge 1\,\,{\mathop{m int}} \,\,\,\,\, \Rightarrow \,\,\,? = 8!\,\,} \hfill \cr

} } ight.

\left( {1 + 2} ight)\,\,\,\,? = 8!\,\,\,\,\, \Rightarrow \,\,\,\,\left( {m{C}} ight)

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

Regards, 
Fabio.

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

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