M is the sum of the reciprocals of the consecutive integers from 201

Question

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

Bunuel · Accepted Answer

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 largest 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 the actual sum is more than that, 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 the actual sum is less than that, we have that M < 1/2.

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

Answer: A.

mike34170 · Answer

I'd handle this differently:

Numbers range from 201 to 300, so the average is approximatively 250.

Then, the sum of all reciprocals equals to 100 times 1/250, i.e. 100/250 ---> 1/2,5

Answer A

BrentGMATPrepNow · Answer

We want to find 1/201 + 1/202 + 1/203 + . . . + 1/299 + 1/300

NOTE : there are 100 fractions in this sum.

Let's examine the extreme values (1/201 and 1/300)

First consider a case where all of the values are equal to the   smallest   fraction (1/300)
We get: 1/300 + 1/300 + 1/300 + ... + 1/300 = 100/300 = 1/3
So, the original sum must be  greater than 1/3

Now consider a case where all of the values are equal to the   biggest   fraction (1/201)
In fact, let's go a little bigger and use 1/200
We get: 1/200 + 1/200 + 1/200 + ... + 1/200 = 100/200 = 1/2
So, the original sum must be  less than 1/2

Combine both cases to get  1/3  < M <  1/2

Answer: A

Cheers,
Brent

ScottTargetTestPrep · Answer

Solution:

Let's first analyze the question. We are trying to find a potential range for M, and M is equal to the sum of the reciprocals from 201 to 300, inclusive. Thus, M is:

1/201 + 1/202 + 1/203 + …+ 1/300

There is no way the GMAT would ever expect us to do this math, and that is exactly why the answer choices in are in the form of an inequality. Thus, we do not need to know the EXACT value of M. The easiest way to determine the RANGE of M is to use easy numbers that can quickly be manipulated.

Note that 1/200 is greater than each of the addends and that 1/300 is less than or equal each of the addends. Therefore, instead of trying to add together 1/201 + 1/202 + 1/203 + …+ 1/300, we are instead going to add 1/200 one hundred times and 1/300 one hundred times. These two sums will give us a high estimate of M and a low estimate of M. Again, we are adding 1/200, one hundred times, and 1/300, one hundred times, because there are 100 numbers from 1/201 to 1/300.

Instead of actually adding each one of these values one hundred times, we will simply multiply each value by 100. We have:

1/300 x 100 = 1/3

1/200 x 100 = 1/2

We see that M is between 1/3 and 1/2.

Answer A.

imperfectdark · Answer

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:

(1/250)*100 = (1/25) * (1/10) * 100 = (1/25) * 10 = (0.04)*10 = 0.40

And by inspecting the answers 1/3 < 0.4 < 1/2 so I get (A).

avigutman · Answer

Video solution from Quant Reasoning:
 https://www.youtube.com/watch?v=lBd0uMp3TwE

fukirua · Answer

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

mbaiseasy · Answer

Whoah! I would never approach that question this way. 
Thank you for this post!

AbhiJ · Answer

Now that you know you would

.

Bunuel · Answer

Similar question to practice: if-k-is-the-sum-of-reciprocals-of-the-consecutive-integers-145365.html Hope it helps

coolparthi · Answer

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.

Bunuel · Answer

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.

raviventrapragada · Answer

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.

harishbiyani · Answer

Wow, that was an awesome explanation. Very helpful.

kshitij89 · Answer

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 1/3 ?
Is the above generalisation correct ?

EMPOWERgmatRichC · Answer

Hi Manonamission,

In this prompt, the answer choices are "ranges"; this usually means that there's a way to avoid doing lots of math and instead use patterns and logic to save you time. You can actually stop working once you figure out the minimum value...

Since 1/300 < 1/201 and the sum of those 100 terms would be (100)(1/300) = 1/3 AT THE MINIMUM, the only answer that's possible would be A. The extra work that you could do to find the maximum value just confirms what the sum would be at that 'level', but but that work is unnecessary.

As you continue to study, be mindful of how the answer choices are written - they can sometimes provide a huge hint into the fastest way to answer the question.

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

saurabhsavant · Answer

Simply awesome solution....how do i reach where u r...Q60

testcracker · Answer

excellent .... :lol:

AnisMURR · Answer

Here is a detailed explanation in video for this question

https://www.youtube.com/watch?v=fmHUtglnMh0

M is the sum of the reciprocals of the consecutive integers from 201

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M is the sum of the reciprocals of the consecutive integers from 201

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