If 1/n(n+1) = 1/n 1/(n+1), then what is the value of 1/(1*2) + 1/(2*

Question

If  \frac{1}{n(n+1)} = \frac{1}{n} – \frac{1}{(n+1)},  then what is the value of  \frac{1}{(1*2)} + \frac{1}{(2*3)} + \frac{1}{(3*4)} + … + \frac{1}{(99*100)} 
 
A.  \frac{1}{100}

B.  \frac{1}{50}

C.  \frac{49}{50}

D.  \frac{99}{100}

E.  \frac{1}{2}

Rabab36 · Answer

PUTTING VALUES WE WILL GET
FOLLOWING SEQUENCE

1-1/2+1/2-1/3+1/3.............. -1/99+1/99-1/100

SOLVING WE GET 
1-1/100
99/100
 OPTION D COPRRECT

lesgooo · Answer

Can someone please give a thorough explanation for this.
Struggling with sequences!

AllisonBell · Answer

Let’s walk through this using an  Understand, Plan, Solve  framework.

Understand : We’re being asked to find the  sum of a long list of fractions , with a formula provided as a tool we might leverage. Note that our answers are also fractions, and they appear in pairs: two with a denominator of 100 and two with a denominator 50. There is also a spread in the answer choices, with two answers close to 1⁄2, on answer close to 1, and two much smaller fractions. This information may be helpful in estimating or common sense eliminations.

Next,  see how the formula applies to the terms : the left-hand side of the equation, 1/n(n+1), matches the setup of each term on the list. For example, if n = 1, 1/n(n+1) matches the first term on the list, 1/(1*2).

Plan : Given how the formula applies to the terms, one option is to rewrite some of the terms using the right-hand side of the formula.  When asked to sum long lists of terms, it is also helpful to write out the first few and last few terms, looking for patterns.  So write out the first few and last few terms using the right-hand side of the formula.  Plan to check for patterns and see if any answers can be eliminated using estimation or common sense .

Solve : The first term becomes 1 - (1⁄2) = 1⁄2.
The second term becomes 1⁄2 - 1⁄3 = 3/6 - 2/6 = 1⁄6.
The third term becomes 1⁄3 - 1⁄4 = 4/12 - 3/12 = 1/12.

Here, we can pause to notice that our starting term is 1⁄2, and the first three terms that we add are smaller than 1⁄2. Since we will only be adding positive terms, we can expect that the answer will be larger than 1⁄2.  That rules out all answers except for D.

You can learn more about common patterns in GMAT problems and fast, efficient alternative approaches through ManhattanPrep’s Free Starter Kit, which includes a Dynamic Question Builder with thorough explanations by our instructors.

Happy studying!

Ally Bell
ManhattanPrep Instructor

If 1/n(n+1) = 1/n 1/(n+1), then what is the value of 1/(12) + 1/(2

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