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# If k is the sum of the reciprocals of the consecutive intege

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If k is the sum of the reciprocals of the consecutive intege [#permalink]  13 Sep 2017, 12:56
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If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ $$\frac{1}{4}$$

❑ $$\frac{1}{3}$$

❑ $$\frac{1}{2}$$

[Reveal] Spoiler: OA
A,B
[Reveal] Spoiler: OA

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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  13 Sep 2017, 18:47
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Sum up 1/60 20 times to get 1/3.
Answer is definitely more than 1/3. Therefore answer is 1/2, as it is the only option greater than 1/3.
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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  24 Oct 2017, 01:35
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If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ \frac{1}{4}

❑ \frac{1}{3}

❑ \frac{1}{2}

Since K is the sum of 1/41 to 1/60 and it is a evenly spaced set, we can easily calculate the sum of set k as follows:

(1/41 + 1/60)/(2) * 20 = 0,41

We multiply by 20 since we have 20 values in our set.

Hence A an B are the correct answers as they are < 0,41

I hope this helped !
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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  25 Apr 2018, 11:20
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S = 1/41 +1/42 + ... + 1/60
There are 40 elements >= 1/60 in that set => S > 20 * 1/60 = 1/3
Since 1/3 > 1/4 => S > 1/4 also

There are 40 elements < 1/40 in that set => S < 20 *1/40 = 1/2 => Eliminate C

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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  24 May 2018, 08:50
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For max value 1/41 * 20 = 0.48
For least value 1/60 * 20 = 1/3

Hence, K will be less than 1/2 but more than 1/3.

Answer: 1/3 and 1/4 are less than k. (A & B)
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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  08 Aug 2018, 22:21
Carcass wrote:

If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ $$\frac{1}{4}$$

❑ $$\frac{1}{3}$$

❑ $$\frac{1}{2}$$

[Reveal] Spoiler: OA
A,B

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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  06 Sep 2020, 20:58
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Carcass wrote:

If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ $$\frac{1}{4}$$

❑ $$\frac{1}{3}$$

❑ $$\frac{1}{2}$$

[Reveal] Spoiler: OA
A,B

Approximation works to solve this question.

We know that:

$$\frac{1}{40} > \frac{1}{41}, \frac{1}{42}, .... ,\frac{1}{49}$$

And:

$$\frac{1}{50} > \frac{1}{51}, \frac{1}{52}, .... ,\frac{1}{59}$$

So we can approximate $$k$$:

$$k = \frac{1}{41} + \frac{1}{42} + ... + \frac{1}{51} + \frac{1}{52} + .... + \frac{1}{60}$$

$$k < \frac{1}{40} + \frac{1}{40} + .... + \frac{1}{50} + \frac{1}{50} + .... + \frac{1}{60}$$

There are 9 numbers between 41 and 49, and ten between 50 and 59, with the lone 60 at the end, so:

$$k < 9 * \frac{1}{40} + 10 * \frac{1}{50} + \frac{1}{60}$$

This simplifies to:

$$k < \frac{9}{40} + \frac{10}{50} + \frac{1}{60}$$

$$k < \frac{9}{40} + \frac{1}{5} + \frac{1}{60}$$

$$\frac{9}{40}$$ is slightly less than $$\frac{10}{40}$$ or $$\frac{1}{4}$$, and $$\frac{1}{60}$$ is less than $$\frac{1}{50}$$, so essentially what we have is:

$$k < \frac{1}{4} + \frac{1}{5} + \frac{1}{50}$$

$$k < 0.25 + 0.2 + 0.02$$

$$k < 0.47$$

Giving us the answers A and B
Re: If k is the sum of the reciprocals of the consecutive intege   [#permalink] 06 Sep 2020, 20:58
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