What is the product of all values generated by the expression

Question

What is the product of all values generated by the expression ( 1-\frac{1}{n^2} ) for the integers  2 \leq n \leq 16  ?

A.  \frac{7}{13}

B.  \frac{15}{28}

C.  \frac{8}{15}

D.  \frac{17}{32}

E.  \frac{9}{17}

Solenja · Answer

hey i did this question but in dumb way im pretty sure there might be shortcut for this could you please explain it.

gmatophobia · Answer

Hi ! I tried to look for a pattern for different values of n. I tried out three values and was able to see a generalized form in the output. Used that generalized form to arrive at the answer. Here is a video of my attempt - https://www.youtube.com/watch?v=wHoOjCYzjbE Hope this helps! P.S - While I was able to solve the question in under 2 mins, the video is longer because of the explanation

Regor60 · Answer

First change the expression to:

(n+1)((n-1)/n^2

Then do the first three:

(2+1)(2-1)/2^2
(3+1)(3-1)/3^2
(4+1)(4-1)/4^2

Notice that the plus and minus of the first and third terms cancels the denominator of the 2nd term.

Notice that for the first term the denominator will only reduce 1 power since there is no preceding term, leaving

1/2

Realize this progression will continue through the 15th term (n=16) with cancelling denominators except there is no 16th term (17-1) to cancel the 2nd power of 16 in the denominator of the 15th term.

So 1/16 remains.

Since there is no 16th term there's also no power of 17 in the denominator to cancel the (16+1) in the numerator of the 15th term.

So 17 remains.

Complete multiplication is then:

1/2*1/16*17 = 17/32

An intuitive way is to realize the question would be posed including many cancellations and that a multiple of 16 is likely to remain due to likely cancellations.

The only answer with a multiple of 16 in the denominator is 17/32.

Posted from my mobile device

JeffTargetTestPrep · Answer

Since 1 – 1/n^2 = (n^2 – 1)/n^2, the product in question is:

( /2^2)( /3^2)( /4^2)…( /16^2)

Using the difference of squares identity, we could write each difference in the numerators as the product of a sum and a difference. Notice that each sum, except for the last, can be canceled out with a corresponding factor in the denominator of the next fraction. So, we have:

( /2^2)( /3)( /4^2)…( /16)

Now, notice that each difference, except for the first, can be canceled out with a corresponding factor in the denominator of the preceding fraction. So, we have:

( /2)( /16) =
(1/2)(17/16) =
17/32

Answer: D

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the product of all values generated by the expression

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the product of all values generated by the expression

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards