If x is an integer greater than 2, the function f(x)

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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

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by [email protected] » Fri Jan 18, 2019 5:26 am
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
The key here is that f(51) is EQUAL to f(50)
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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by [email protected] » Fri Jan 18, 2019 5:14 pm
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
$$f\left( x \right) = \left\{ \matrix{
\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot x\,\,\,\,,\,\,\,\,x\,\,{\rm{even}}\,\,\,\,\left( * \right) \hfill \cr
\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot \left( {x - 1} \right)\,\,\,\,,\,\,\,\,x\,\,{\rm{odd}}\,\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,x \ge 3\,\,{\mathop{\rm int}} \,} \right]$$
$$? = f\left( {51} \right) - f\left( {50} \right)$$
$$f\left( {51} \right)\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,2 \cdot 4 \cdot 6 \cdot \ldots \cdot \left( {51 - 1} \right)\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,f\left( {50} \right)\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0$$


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DivyaD wrote:
Fri Jan 18, 2019 4:17 am
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
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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Junior | Next Rank: 30 Posts
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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)

Junior | Next Rank: 30 Posts
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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