P is the product, indicated above, of all the numbers of the form

Question

P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})

P  is the product, indicated above, of all the numbers of the form  1 - \frac{1}{k}  where  k  is an integer from 2 to 50, inclusive. What is the value of  P ?

A)  \frac{1}{100}

B)  \frac{1}{50}

C)  \frac{1}{49}

D)  \frac{49}{50}

E)  \frac{99}{100}

Himanshu9818 · Accepted Answer

P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})

P  is the product, indicated above, of all the numbers of the form  1 - \frac{1}{k}  where  k  is an integer from 2 to 50, inclusive. What is the value of  P ?

A) 1/100

B) 1/50

C) 1/49

D) 49/50

E) 99/100

Ans B.
File attached

Abhishek009 · Answer

Dear Sallyzodiac and Himanshu9818

There is a pattern if you all can analyze -

Sallyzodiac

\(P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})\)

\(P = (1-\frac{1}{2})\) = \(\frac{1}{2}\)

\(P = (1-\frac{1}{2})(1-\frac{1}{3})\) = \((\frac{1}{3})\)

\(P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})\) = \((\frac{1}{4})\)

Can you notice a pattern ?

I think by now all of you have understood that it depends on the value of the last denominator...

So, \(P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})\) = \((\frac{1}{50})\)

Hence answer will be (B) ,\((\frac{1}{50})\)

Nathanlambson · Answer

So does this mean that if k was from 2 to 10 inclusive, the answer would be 1/10? I'm having a hard time following the math on this one.

JeffTargetTestPrep · Answer

Let’s calculate some values from our set to find a pattern:

1 - ½ = ½

1 - ⅓ = ⅔

1 - ¼ = ¾

1 - ⅕ = ⅘

Multiplying these values together, we have:

(1/2)(2/3)(3/4)(4/5)

Notice that the fractions simplify and we are left with ⅕.

So, we are left with the numerator from the first fraction and the denominator from the last fraction.

For P, since the last fraction in the set is 1 - 1/50 = 49/50, P would equal 1/50.

Answer: B

generis · Answer

Nathanlambson  - yes, if k were 2 to 10 inclusive, the answer would be 1/10.

I had a hard time following the math, too. Perhaps how I rewrote the post might help.

1. From original equation, instead of considering all the factors in parentheses, take them  two at a time , and do any arithmetic that is inside parentheses.

P = (1-\frac{1}{2})(1-\frac{1}{3})  =

(\frac{1}{2})(\frac{2}{3})  =  \frac{2}{6}  =   \frac{1}{3}

2. Repeat, but put the end product from above as the first of your next two factors

(\frac{1}{3})(1-\frac{1}{4})  =  (\frac{1}{3})(\frac{3}{4})  =  \frac{1}{4}

And again, two at a time, use end product from previous step as first factor:

(\frac{1}{4})(1-\frac{1}{5})  =  (\frac{1}{4})(\frac{4}{5})  =  \frac{1}{5}

Abhishek009  wrote   "  depends on the value of the last denominator...  .

In the last step above, denominator is 5. 1 - (1/ 5 ) = 4/ 5

The pattern also demonstrates that after multiplication by previous term, the numerator is 1, and the product's denominator is 5.  (\frac{1}{4})(\frac{4}{5})  =  \frac{1}{5} .

So the pattern is: the end product of all factors at any point including last term = 1 over whatever denominator the nth term has.

You could calculate k = 2 to 10 inclusive, easily, by multiplying  only  the first two factors, and by inserting the answer to the arithmetic inside all the parentheses. Thus:

(1-\frac{1}{2})(1-\frac{1}{3}) 
= (\frac{1}{2})(\frac{2}{3})  =  \frac{1}{3}  ...

\frac{1}{3} * \frac{3}{4} * \frac{4}{5}  * \frac{5}{6} * \frac{6}{7} *  \frac{7}{8}  * \frac{8}{9} * \frac{9}{10}  =

Start canceling. You get  \frac{1}{10}

Hope it helps.

Boozer · Answer

Another way to look at it is to turn P into a factorial.

(1)   1 - \frac{1}{2} = \frac{1}{2}

(2)   1 - \frac{1}{3} = \frac{2}{3}

(3)   1 - \frac{1}{4} = \frac{3}{4}

(4)   1 - \frac{1}{5} = \frac{4}{5}

P=\frac{1}{2} * \frac{2}{3} * \frac{3}{4} * \frac{4}{5} * . . . * \frac{49}{50}

∴  P=\frac{49!}{50!} = \frac{49!}{50*49!} = \frac{1}{50}

Answer  B .

1MeowMeowMeow1 · Answer

... .... ....... ........ .......

Bunuel · Answer

___________________________
Tagged as GMAT Prep. Thank you.

YuktiMaheshwari · Answer

(1-1/2)*(1-1/3)….(1-1/50)
1/2*2/3*…….49/50

1*2*3………*49
_____________ = 1/50 (B)
2*3*……….*50

Posted from my mobile device

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P is the product, indicated above, of all the numbers of the form

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