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Question of the Week - 42 (The value of the variable E is determined)

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Question of the Week - 42 (The value of the variable E is determined)  [#permalink]

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04 Apr 2019, 22:05
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Difficulty:

65% (hard)

Question Stats:

52% (02:15) correct 48% (01:52) wrong based on 27 sessions

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Question of the Week #42

The value of the variable E is determined by adding the reciprocals of the first 10 even natural numbers. Which of the following can be a possible value of the reciprocal of E?

A. 0.154
B. 0.1818
C. 0.667
D. 2
E. 3.03

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Re: Question of the Week - 42 (The value of the variable E is determined)  [#permalink]

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05 Apr 2019, 02:20
Given
E= determined by adding the reciprocals of the first 10 even natural numbers ; 2,4,6,8,10..20
so E = (1/2+1/4+1/6+1/8+1/10+1/12+...+1/20)
or say
E = 1/2(1/1+1/2+1/3+1/4+1/5+1/6+...+1/10)
E= 1/2 ( 1+.5+.3+.25+.2+.16+....+.1)
E= ~ 3/2
reciprocal of E = 2/3 = ~ 0.66
IMO C

EgmatQuantExpert wrote:
Question of the Week #42

The value of the variable E is determined by adding the reciprocals of the first 10 even natural numbers. Which of the following can be a possible value of the reciprocal of E?

A. 0.154
B. 0.1818
C. 0.667
D. 2
E. 3.03

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Joined: 24 Nov 2018
Posts: 107
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Re: Question of the Week - 42 (The value of the variable E is determined)  [#permalink]

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05 Apr 2019, 17:07
EgmatQuantExpert wrote:
Question of the Week #42

The value of the variable E is determined by adding the reciprocals of the first 10 even natural numbers. Which of the following can be a possible value of the reciprocal of E?

A. 0.154
B. 0.1818
C. 0.667
D. 2
E. 3.03

Sum of first 10 even natural numbers=$$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+.....+\frac{1}{20}=x$$
applying Geometric Progression,$$x= 1/2(1-(1/2^10))/(1-1/2)=1023/1024$$
x<1 and >0.5, Only option satisfying that is C. IMo, option C.
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Re: Question of the Week - 42 (The value of the variable E is determined)  [#permalink]

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07 Apr 2019, 06:42
Top Contributor
EgmatQuantExpert wrote:
The value of the variable E is determined by adding the reciprocals of the first 10 even natural numbers. Which of the following can be a possible value of the reciprocal of E?

A. 0.154
B. 0.1818
C. 0.667
D. 2
E. 3.03

A fast approach is to use some estimation.

E = 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14 + 1/16 + 1/18 + 1/20
≈ 0.5 + 0.25 + 0.15 + 0.1 + 0.1 + 0.1 + 0.05 + 0.05 + 0.05 + 0.05
1.4

Which of the following can be a possible value of the reciprocal of E?
The reciprocal of E ≈1/1.4 ≈0.7

Check the answer choices . . . answer = C

Cheers,
Brent
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Question of the Week - 42 (The value of the variable E is determined)  [#permalink]

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10 Apr 2019, 21:40

Solution

Given:

In this question, we are given that
• The value of the variable E is determined by adding the reciprocals of the first 10 even natural numbers.

To find:

We need to determine
• Among the given options, which one can be a possible value of the reciprocal of E.

Approach and Working:

As E equals the sum of the reciprocals of the first 10 even natural numbers, we can write E in the following manner:
• E = $$\frac{1}{2}$$ + $$\frac{1}{4}$$ + $$\frac{1}{6}$$ + $$\frac{1}{8}$$ + $$\frac{1}{10}$$ + $$\frac{1}{12}$$ + $$\frac{1}{14}$$ + $$\frac{1}{16}$$ + $$\frac{1}{18}$$ + $$\frac{1}{20}$$

Or, E = $$\frac{1}{2} (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10})$$

Now, if we observe the terms within the bracket carefully, we can see that 1 is the largest among all the 10 terms and $$\frac{1}{10}$$ is the smallest among all the 10 terms.
• If all 10 terms were equal to $$\frac{1}{10}$$, then E would be $$\frac{1}{2} * 10 * \frac{1}{10}$$ = $$\frac{1}{2}$$.
o But since the actual terms are more, we can say that the value of E is greater than $$\frac{1}{2}$$.

• Similarly, if all 10 terms were equal to 1, then E would be $$\frac{1}{2} * 10 * 1 = 5$$.
o But since the actual terms are less, we can say that the value of E is less than 5.

Combining the above results, we can say
• $$\frac{1}{2} < E < 5$$
Or, $$2 > \frac{1}{E} > \frac{1}{5}$$
Or, $$2 > \frac{1}{E} > 0.2$$

From the given options, only 0.667 falls in the given range.

Hence, the correct answer is option C.

Answer: C

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Question of the Week - 42 (The value of the variable E is determined)   [#permalink] 10 Apr 2019, 21:40
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Question of the Week - 42 (The value of the variable E is determined)

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