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# If S is the sum of the reciprocals of the consecutive

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If S is the sum of the reciprocals of the consecutive [#permalink]

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23 Dec 2005, 02:19
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If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III
[Reveal] Spoiler: OA

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JAI HIND!

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23 Dec 2005, 02:47
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I think it's C, but I'm not quite sure.

Since we summarize the reciprocals from 100 to 91, we can say also that we add ten numbers who are all (with one exception 1/100) greater than 1/100, so that the sum must be greater than 1/10.

On the other side we can say that we add the reciprocals from 91 to 100, so that the sum has to be less than the sum of ten times 1/91.

We can conclude that the sum has to be less than 1/9 but more than 1/10. That leaves us C as the only possible answer.
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23 Dec 2005, 03:14
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JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

S = 1/91 + 1/92 + 1/93.....+ 1/100
S = (1/91 + 1/100)(10/2) = 191/1820 = approx 0.1049.....

I. 1/8 = 0.12...
II. 1/9 = 0.11....
III. 1/10 = 0.10
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23 Dec 2005, 05:26
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JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

From the answer choice we get 1) 1/8 = .125
2) 1/9 = .11 (Approx)
3) 1/10 = .1

1/100 = .01 The other numbers from 1/91 to 1/99 will also be close to .01 but little greater than that . so approx .01 * 10 = .1 So its C.
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23 Dec 2005, 23:44
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This question is a repetition here.
Anyways lemme explain the solution

1/91+1/92+..........1/100 > 1/100+1/100......10 times
or Summation S >10/100
i.e S>1/10

Similarly
1/91+1/92+..........1/100 < 1/91+1/91......10 times
Summation < 10/91 <1/9

Hence summation is only greater than 1/10.........Hence C
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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25 Nov 2013, 08:12
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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25 Nov 2013, 08:43
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JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

Given that $$S=\frac{1}{91}+\frac{1}{92}+\frac{1}{93}+\frac{1}{94}+\frac{1}{95}+\frac{1}{96}+\frac{1}{97}+\frac{1}{98}+\frac{1}{99}+\frac{1}{100}$$. Notice that 1/91 is the larges term and 1/100 is the smallest term.

If all 10 terms were equal to 1/91, then the sum would be 10/91, but since actual sum is less than that, then we have that S<1/91.

If all 10 terms were equal to 1/100, then the sum would be 10/100=1/10, but since actual sum is more than that, then we have that S>1/10.

Therefore, 1/10 < S < 10/91.

Also, notice that 10/91 < 1/9 < 1/8, thus we have that 1/10 < S < 10/91 < 1/9 < 1/8.

Therefore only 1/10 is less than S.

Similar questions to practice:
m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html
if-k-is-the-sum-of-reciprocals-of-the-consecutive-integers-145365.html
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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10 Jan 2014, 11:40
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JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

We know that there are 10 numbers in the sum: (100-91)+1=10
Take the mean of the sum and times it by 10 to get our sum: (1/ ((100+91)/2)) x 10 = 10/95.5 = 1/9.55

From here we know that the only number which will be smaller than our sum must be divisible by >9.55

Hence only III satisfies. Answer: C
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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20 Feb 2014, 16:30
some person solved this with an Arithmetic progression formula, is that possible, or just a coincidence?
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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21 Feb 2015, 23:20
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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21 Feb 2015, 23:50
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Hi All,

The other explanations in this thread have properly explained the "math" behind this prompt - it's essentially about figuring out the "minimum" and "maximum" value that the sum COULD be, then realizing the sum is between those two values. Any time you find yourself reading a Quant question and you think "the math will take forever", then you're probably right AND there should be another way to get to the correct answer. The Quant section of the GMAT is NOT a "math test" (at least not in the way that you might be used to thinking about it). Yes, you will do plenty of small calculations and use formulas, but the Quant section is there to test you on LOTS of other non-math related skills: organization, accuracy, attention to detail, ability to prove that you're correct, pattern-matching, pacing, etc. To maximize your performance on Test Day, you have to be a stronger 'strategist' and 'pattern-matcher' than 'mathematician.'

GMAT assassins aren't born, they're made,
Rich
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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22 Mar 2015, 12:10
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

S = 1/91 + 1/92 + ... + 1/100

or S > 1/100 + 1/100 +... + 1/100
S > 1/10

or S < 10/91
S < 0.109

=> 0.1 < S < 0.109

hence C.
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Re: If S is the sum of the reciprocals of the consecutive [#permalink]

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20 Jan 2016, 18:30
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Re: If S is the sum of the reciprocals of the consecutive   [#permalink] 20 Jan 2016, 18:30
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