If x is an integer greater than 2, the function f(x) represents the product of all even integers between 2 and x, inclusive. What is f(51) f(50)?
a) (51)50!
b) (502)49!
c) 50
d) 1
e) 0
If x is an integer greater than 2, the function f(x)
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The key here is that f(51) is EQUAL to f(50)DivyaD wrote:If x is an integer greater than 2, the function f(x) represents the product of all even integers between 2 and x, inclusive. What is f(51)  f(50)?
a) (51)50!
b) (502)49!
c) 50
d) 1
e) 0
Here's why:
According to the definition of the function f, f(51) = (2)(4)(6). . . (48)(50), and f(50) = (2)(4)(6). . . (48)(50)
So, f(51)  f(50) = (2)(4)(6). . . (48)(50)  (2)(4)(6). . . (48)(50)
= 0
Answer: E
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$$f\left( x \right) = \left\{ \matrix{DivyaD wrote:If x is an integer greater than 2, the function f(x) represents the product of all even integers between 2 and x, inclusive. What is f(51) f(50)?
a) (51)50!
b) (502)49!
c) 50
d) 1
e) 0
\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot x\,\,\,\,,\,\,\,\,x\,\,{\rm{even}}\,\,\,\,\left( * \right) \hfill \cr
\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot \left( {x  1} \right)\,\,\,\,,\,\,\,\,x\,\,{\rm{odd}}\,\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,x \ge 3\,\,{\mathop{\rm int}} \,} \right]$$
$$? = f\left( {51} \right)  f\left( {50} \right)$$
$$f\left( {51} \right)\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot \left( {51  1} \right)\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,f\left( {50} \right)\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0$$
We follow the notations and rationale taught in the GMATH method.
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Since both f(51) and f(50) are the product of all even integers from 2 to 50, inclusive, the difference between f(51) and f(50) is 0.
Answer: E
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Let us evaluate f(51) in terms of f(50).
Since f(51) = 2 * 4 * 6 * ... * 50
And f(50) = 2 * 4 * 6 * ... * 50
Therefore f(51) = f(50)
So f(51)  f(50) = f(50)  f(50) = 0 (E)
Since f(51) = 2 * 4 * 6 * ... * 50
And f(50) = 2 * 4 * 6 * ... * 50
Therefore f(51) = f(50)
So f(51)  f(50) = f(50)  f(50) = 0 (E)
Given,
x > 2
f(x) represents the product of all even integers between 2 and x, inclusive.
f(51) = 2 * 4 * 6 * 8 *.............................*48 * 50
f(50) = 2 * 4 * 6 * 8 *.............................*48 * 50
So, f(51)  f(50) = 0
Answer E
x > 2
f(x) represents the product of all even integers between 2 and x, inclusive.
f(51) = 2 * 4 * 6 * 8 *.............................*48 * 50
f(50) = 2 * 4 * 6 * 8 *.............................*48 * 50
So, f(51)  f(50) = 0
Answer E