For all positive integers r, the sequence Sr is given by the formula

Question

For all positive integers  r , the sequence  S_r  is given by the formula 
 S_r = \frac{{1}}{{r+1}}-\frac{1}{r}  
What is the sum of the first 20 terms of the sequence?

(A)  \frac{-22}{21}

(B)  \frac{-20}{21}

(C)  \frac{1}{21}

(D)  \frac{20}{21}

(E)  \frac{22}{21}

GMATNinja · Answer

Check out question 6 in the video below for the solution:

https://www.youtube.com/watch?v=LkDq3hwlUMw&t=2784s

MS61 · Answer

What is the approach to this one ?

Bunuel · Answer

For all positive integers  r , the sequence  S_r  is given by the formula    S_r = \frac{{1}}{{r+1}}-\frac{1}{r}

What is the sum of the first 20 terms of the sequence?

(A)  \frac{-22}{21}

(B)  \frac{-20}{21}

(C)  \frac{1}{21}

(D)  \frac{20}{21}

(E)  \frac{22}{21} 
 
You can check GMATNinja's explaination in the video above. However, here it is again.

Since  S_r = \frac{{1}}{{r+1}}-\frac{1}{r} , then:
 
 
 s_1 = \frac{1}{1+1} - \frac{1}{1} = \frac{1}{2} - 1

s_2 = \frac{1}{2+1} - \frac{1}{2} = \frac{1}{3} - \frac{1}{2}

s_3 = \frac{1}{3+1} - \frac{1}{3} = \frac{1}{4} - \frac{1}{3}

...

s_{19} = \frac{1}{19+1} - \frac{1}{19} = \frac{1}{20} - \frac{1}{19}

s_{20} = \frac{1}{20+1} - \frac{1}{20} = \frac{1}{21} - \frac{1}{20}

Therefore, the sum of the first 20 terms of the sequence is:

( \frac{1}{2} - 1) + ( \frac{1}{3} - \frac{1}{2}) + (\frac{1}{4} - \frac{1}{3}) + ... + (\frac{1}{20} - \frac{1}{19}) + (\frac{1}{21} - \frac{1}{20})=

= - 1 + \frac{1}{21} =

=- \frac{20}{21}

Answer: B.

Kinshook · Answer

@GMATNinjaGiven: For all positive integers  r , the sequence  S_r  is given by the formula 
 S_r = \frac{{1}}{{r+1}}-\frac{1}{r}  
Asked: What is the sum of the first 20 terms of the sequence?
The first 20 terms of the sequence =  S_1 + S_2 + ..... + S_{20} = (\frac{1}{2} - \frac{1}{1}) + (\frac{1}{3} - \frac{1}{2}) + .. . + (\frac{1}{21} - \frac{1}{20}) = \frac{1}{21} - \frac{1}{1} = \frac{(1 - 21)}{21} = -\frac{20}{21}

​​​​​​​IMO B

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For all positive integers r, the sequence Sr is given by the formula

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For all positive integers r, the sequence Sr is given by the formula