Sum of the fractions

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sat May 03, 2014 10:42 am

Hello,

Can you please tell me how to solve this:

What is 1/(1)(2) + 1/(2)(3) + 1/(3)(4) + 1/(4)(5) + 1/(5)(6) + 1/(6)(7) + 1/(7)(8) + 1/(8)(9) + 1/(9)(10) ?

OA: [spoiler]9/10 [/spoiler]

Thanks a lot,
Sri

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Source: Beat The GMAT — Problem Solving | Sat May 03, 2014 10:42 am

raj44: Senior | Next Rank: 100 Posts; Posts: 39; Joined: Sat Mar 15, 2014 10:36 pm; Thanked: 1 times

by raj44 » Sat May 03, 2014 11:24 pm

Hi,

This can be solved like :

1. Determine the nth term of the series which happens to be 1/(n)(n+1) where 1<= n <= 9

2. Now we can employ partial fraction skills to separate the multiplying denominators and determine the Numerator coefficients:

therefore, the exp 1/(n)(n+1)= 1/n-1/n+1 where 1<= n <= 9.

This is called a telescoping series. write out the first few terms of each of these two series:

1/n: 1, 1/2, 1/3, 1/4, 1/5, ...

-1/(n+1): -1/2, -1/3, -1/4, -1/5, -1/6, ...

Add these together and you should notice that almost everything cancels except for a 1 at the beginning of the 1/n series and a term at the end of the -1/(n+1) series.

Therefore we have 1-0.1=0.9= 9/10 as the answer.

gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

What is 1/(1)(2) + 1/(2)(3) + 1/(3)(4) + 1/(4)(5) + 1/(5)(6) + 1/(6)(7) + 1/(7)(8) + 1/(8)(9) + 1/(9)(10) ?

OA: [spoiler]9/10 [/spoiler]

Thanks a lot,
Sri

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Sun May 04, 2014 12:03 am

Hi Sri,

What was the original question (and answers) that this concept was a part of?

In many cases, a Quant question that looks like it requires "lots of math" is actually based on a pattern that will allow you to avoid most of that math. Sometimes the answers are written in such a way that you can also avoid most of the math.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun May 04, 2014 2:21 am

gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

What is 1/(1)(2) + 1/(2)(3) + 1/(3)(4) + 1/(4)(5) + 1/(5)(6) + 1/(6)(7) + 1/(7)(8) + 1/(8)(9) + 1/(9)(10) ?

OA: [spoiler]9/10 [/spoiler]

Thanks a lot,
Sri

If x = 1/2 + 1/6 + 1/12 + 1/20 + ... + 1/90, what is the value of x?
WRITE IT OUT and LOOK FOR A PATTERN.

Sum of the first 2 terms:
1/2 + 1/6 = 2/3.

Sum of the first 3 terms:
2/3 + 1/12 = 3/4.

Sum of the first 4 terms:
3/4 + 1/20 = 4/5.

In each case:
The NUMERATOR of the sum = the NUMBER OF TERMS.
THE DENOMINATOR of the sum = NUMERATOR + 1.

Thus:
x = sum of the first 9 terms = [spoiler]9/10[/spoiler].

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Mon May 05, 2014 5:49 pm

GMATGuruNY wrote:
gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

What is 1/(1)(2) + 1/(2)(3) + 1/(3)(4) + 1/(4)(5) + 1/(5)(6) + 1/(6)(7) + 1/(7)(8) + 1/(8)(9) + 1/(9)(10) ?

OA: [spoiler]9/10 [/spoiler]

Thanks a lot,
Sri
If x = 1/2 + 1/6 + 1/12 + 1/20 + ... + 1/90, what is the value of x?
WRITE IT OUT and LOOK FOR A PATTERN.

Sum of the first 2 terms:
1/2 + 1/6 = 2/3.

Sum of the first 3 terms:
2/3 + 1/12 = 3/4.

Sum of the first 4 terms:
3/4 + 1/20 = 4/5.

In each case:
The NUMERATOR of the sum = the NUMBER OF TERMS.
THE DENOMINATOR of the sum = NUMERATOR + 1.

Thus:
x = sum of the first 9 terms = [spoiler]9/10[/spoiler].

Hello Mitch,

Thanks a lot for the detailed solution.

Best Regards,
Sri

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