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Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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11 Dec 2012, 10:02

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Bunuel wrote:

M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

(A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9

Given that \(M=\frac{1}{201}+\frac{1}{202}+\frac{1}{203}+...+\frac{1}{300}\). Notice that 1/201 is the larges term and 1/300 is the smallest term.

If all 100 terms were equal to 1/300, then the sum would be 100/300=1/3, but since actual sum is more than that, then we have that M>1/3.

If all 100 terms were equal to 1/200, then the sum would be 100/200=1/2, but since actual sum is less than that, then we have that M<1/2.

Therefore, 1/3<M<1/2.

Answer: A.

WOW, that is elegant! Bunuel please suggest if we can use the next assumption: Arithmetic mean of the elements a1 and a100= 501/(300*2*201). Sum of all elements = 100*Arithmetic mean=167/(6*67)= 167/402 , which is definitely more than 1/3. A Thanks
_________________

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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05 Sep 2013, 23:44

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I did this similar to the post above.

Between 201 and 300 there are 100 numbers (300-201+1 = 100). Since the integers 201 to 300 are consecutive, the middle number is ((201+300)/2) which is roughly 250. Therefore, the sum is approximately:

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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28 Nov 2013, 20:21

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Walkabout wrote:

M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

(A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9

There is also an another easier way to solve this problem - using sum and average concept. 1. The numbers are reciprocal of consecutive numbers. Total number of items = 100 (201-300 +1 ) => Middle term is 250 and the number is the reciprocal which is 1/250.

2. Now we know for consecutive numbers and odd number of terms, Average = Middle number (also Average = Median) and Average in general is Sum / Number of terms. In this case average = 1/250.

Hence, we have Average (1/250) = Sum / 100. Solving this sum = 100/250 = 0.4

From the answer choices, 0.4 is between 1/3 and 1/5.

M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

(A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9

There is also an another easier way to solve this problem - using sum and average concept. 1. The numbers are reciprocal of consecutive numbers. Total number of items = 100 (201-300 +1 ) => Middle term is 250 and the number is the reciprocal which is 1/250.

2. Now we know for consecutive numbers and odd number of terms, Average = Middle number (also Average = Median) and Average in general is Sum / Number of terms. In this case average = 1/250.

Hence, we have Average (1/250) = Sum / 100. Solving this sum = 100/250 = 0.4

From the answer choices, 0.4 is between 1/3 and 1/5.

First of all the average of consecutive integers from 201 to 300, inclusive is (201+300)/2=250.5 not 250.

Next, set {1/201, 1/202, 1/203, ..., 1/300} is NOT evenly spaced, thus your method is an approximation (the formula you apply is for an evenly spaced set).

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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29 Nov 2013, 11:45

Bunuel wrote:

coolparthi wrote:

Walkabout wrote:

M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

(A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9

There is also an another easier way to solve this problem - using sum and average concept. 1. The numbers are reciprocal of consecutive numbers. Total number of items = 100 (201-300 +1 ) => Middle term is 250 and the number is the reciprocal which is 1/250.

2. Now we know for consecutive numbers and odd number of terms, Average = Middle number (also Average = Median) and Average in general is Sum / Number of terms. In this case average = 1/250.

Hence, we have Average (1/250) = Sum / 100. Solving this sum = 100/250 = 0.4

From the answer choices, 0.4 is between 1/3 and 1/5.

First of all the average of consecutive integers from 201 to 300, inclusive is (201+300)/2=250.5 not 250.

Next, set {1/201, 1/202, 1/203, ..., 1/300} is NOT evenly spaced, thus your method is an approximation (the formula you apply is for an evenly spaced set).

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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30 Nov 2013, 22:47

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There is another simple way to do this problem, when reciprocals of numbers from 201 to 300 are added...then it means the denominators are in Arithmetic progression...so 1/201, 1/202, ---, 1/300 are in Harmonic progression. Sum of all these numbers is (# of terms)*(2*First term*Last term)/(First term+Last term) which is (2*(1/201)*(1/300))/(1/201+1/300)...simplifying gives us 200/501 ~ 200/500 = 1/2.5 so lies between 1/3 and 1/2 Ans A.

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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02 Jan 2014, 08:14

I will go with percentage If all are 1/201 then (1/201)*100=close to 1/2% so sum= (1/2)*100 =50% Again, if all are 1/300 then (1/300)*100= 1/3% so sum = (1/3)*100 =33.33%

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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07 Jan 2014, 19:05

Bunuel wrote:

coolparthi wrote:

Walkabout wrote:

M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

(A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9

There is also an another easier way to solve this problem - using sum and average concept. 1. The numbers are reciprocal of consecutive numbers. Total number of items = 100 (201-300 +1 ) => Middle term is 250 and the number is the reciprocal which is 1/250.

2. Now we know for consecutive numbers and odd number of terms, Average = Middle number (also Average = Median) and Average in general is Sum / Number of terms. In this case average = 1/250.

Hence, we have Average (1/250) = Sum / 100. Solving this sum = 100/250 = 0.4

From the answer choices, 0.4 is between 1/3 and 1/5.

First of all the average of consecutive integers from 201 to 300, inclusive is (201+300)/2=250.5 not 250.

Next, set {1/201, 1/202, 1/203, ..., 1/300} is NOT evenly spaced, thus your method is an approximation (the formula you apply is for an evenly spaced set).

Hope it's clear.

Is there a reason why the "evenly spaced set" strategy works for this problem but wouldn't work for other problems? For example, what if the question asked for the sum of the consecutive odd reciprocals, or the sum of the reciprocals of consecutive multiples of 7? Would you use the same strategy you mentioned and just set the minimum to (100)(1/207)?

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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11 Jun 2014, 02:53

I thought about it this way.

First we know that 1/201.....to 1/300 = 100 terms. And that in fractions the higher the denominator value relative to the numerator the smaller the fraction. i.e because 1 < 201. therefore 1/200 > 1/201....or for that matter the sum of terms 1/201 to 1/300.

So 1/200 * 100 you get M must be <1/2.

Take a quick glance at the answer choices and you see that only A addresses this fact.

Choose A and move along.

The question is testing your ability to recognize that the higher the denominator value the smaller the number once you recognize this the answer becomes clear

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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21 Jun 2014, 05:00

Will the answer pattern remain the same for the varies interval nos. ?

Suppose , if M is the sum of reciprocals of the cons. integers from 301 to 400 then the answer will be 1/4[m]1/3 ? Is the above generalisation correct ?

Re: M is the sum of the reciprocals of the consecutive integers [#permalink]

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22 Jun 2014, 00:36

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kshitij89 wrote:

Will the answer pattern remain the same for the varies interval nos. ?

Suppose , if M is the sum of reciprocals of the cons. integers from 301 to 400 then the answer will be 1/4[m]1/3 ? Is the above generalisation correct ?

your generalization would be correct but as you notice this form of pattern matching is a quick way to get to the right answer when you are short of time AND yes this holds true for 1/101 to 200, 1/201 to 300, 1/301 to 400 .......etc. Hope this helps.