Question of the Week - 3 (The value of the variable E is...)

e-GMAT 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

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 = 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14 + 1/16 + 1/18 + 1/20
  Or, E = 1/2 (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 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 1/10 is the smallest among all the 10 terms.

• • If all 10 terms were equal to 1/10, then E would be 1/2 x 10 x 1/10 = 1/2.
  • o But since the actual terms are more, we can say that the value of E is greater than 1/2.
  • Similarly, if all 10 terms were equal to 1, then E would be 1/2 x 10 x 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

• • 1/2 < E < 5
  Or, 2 > 1/E > 1/5
  Or, 2 > 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