What is the value of 1/(1*2*3)+1/(2*3*4)+...+1/(48*49*50)?

Question

What is the value of  A=\frac{1}{1 	imes 2 	imes 3}+\frac{1}{2 	imes 3 	imes 4}+...+\frac{1}{48 	imes 49 	imes 50} ?

A.  \frac{306}{1225}

B.  \frac{637}{2550}

C.  \frac{1175}{4704}

D.  \frac{1199}{4800}

E.  \frac{2497}{9996}

BrushMyQuant · Accepted Answer

In these kind of questions our strategy will always be to break down each term into difference of two terms, such that alternate terms cancel out. In thsi case we can do it as follows

A=\frac{1}{2} * \frac{2}{1 	imes 2 	imes 3}+\frac{2}{2 	imes 3 	imes 4}+...+\frac{2}{48 	imes 49 	imes 50} 
 A=\frac{1}{2} * \frac{3-1}{1 	imes 2 	imes 3}+\frac{4-2}{2 	imes 3 	imes 4}+...+\frac{50-48}{48 	imes 49 	imes 50} 
 A=\frac{1}{2} * ( \frac{3}{1 	imes 2 	imes 3} - \frac{1}{1 	imes 2 	imes 3} + \frac{4}{2 	imes 3 	imes 4} - \frac{2}{2 	imes 3 	imes 4}+...+\frac{50}{48 	imes 49 	imes 50} - \frac{48}{48 	imes 49 	imes 50} ) 
 A=\frac{1}{2} * ( \frac{1}{1 	imes 2 } - \frac{1}{2 	imes 3} + \frac{1}{2 	imes 3 } - \frac{1}{3 	imes 4}+...+\frac{1}{48 	imes 49} - \frac{1}{49 	imes 50} ) 
Now alternate terms will cancel out and we will be left with only two terms
 A=\frac{1}{2} * ( \frac{1}{1 	imes 2} - \frac{1}{49 	imes 50} ) 
 A=\frac{1}{2} * ( \frac{1225-1}{49 	imes 50} ) 
 A=\frac{1}{2} * ( \frac{1224}{49 	imes 50} ) 
 A=\frac{306}{1225}

So, answer will be A
Hope it helps!

georgekaterji · Answer

I think this is a 700 level question, its not easy to come up with a quick strategy where you can solve such a problem in almost 2.5 minutes

koenh · Answer

For questions such as these with long multiplication or lots of arithmetic I prefer to use approximation.

A = 1/1*2*3 + 1/2*3*4 + 1/3*4*5 .. + .. 1/48*49*50
A = 1/6 + 1/24 + 1/60 ...
A = 5/24 + 1/60 ...

Now 5/24 > 4/24 > 1/6. Since the remaining numbers to be added are all relatively small I think it's fair to assume that our answer will be slightly larger than 1/6 but smaller than 1/4. So 1/6 < x < 1/4.

Out of the available answer choices the only one that comes close is 306/1225.

Answer --> Option A.

bazu · Answer

But 5/24 is already > than 1/5, not smaller

KarishmaB · Answer

This is not a GMAT-type question.

msing454 · Answer

There's a similar question, based on exact same concept, in GMAT FE OG Mock #5: https://gmatclub.com/forum/given-that-1 ... 23026.html So this does, indeed, qualify as a GMAT type question.

WiziusCareers1 · Answer

The general term is Tn = 1/(n(n + 1)(n + 2))

{1/(n(n + 1))} - {1/(n + 1)(n + 2)} = {(n + 2) - n}/{n(n + 1)(n + 2)} = 2/{n(n + 1)(n + 2)}

Hence, Tn = 1/2{(1/(n(n + 1)) - 1/(n + 1)(n + 2)}

So, we can write A = 1/2{((1/(1 * 2) - 1/(2 * 3)) + (1/(2 * 3) - 1(3 * 4)) + ............ + (1/(48 * 49) - (1/49 * 50))
 = 1/2{1/(1 * 2) - 1/(49 * 50)}
 = 1/2{1/2 - 1/2450}
 = 612/2450
 = 306/1225

Hence, the answer is (A)

willykuo · Answer

every option is less than 1/4 (actually it's quite closed to but less than 1/4)

Adit_ · Answer

You need to solve this the proper way. You can't solve it just like that using the options, you are right.
All the values are virtually the same in fact if we round it off.
One needs to come up with the proper telescopic series to solve it.
This is basically a nested form of 1/1*2 + 1/2*3...
We need to apply the concept to solve above, twice and they cancel out.

What is the value of 1/(123)+1/(234)+...+1/(484950)?

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