Question

In the given question, there are two statements marked I and II. Decide which of the statements are sufficient to answer the question. Choose your answer from the given alternative.

Is the average of n consecutive integers equal to $1$?
(1) $n$ is even.
(2) If $5$ is the sum of the n consecutive integers, then $0<5<n$

• A. Statement I alone is sufficient to answer the problem.
• B. Statement II alone is sufficient to answer the problem.
• C. Statement I and II both are needed.
• D. Statement I and II both are not sufficient.
• E. Either of the statements I or II is sufficient.

Answer 1

The correct option is E Either of the statements I or II is sufficient.

(1) SUFFICIENT: Statement (1) states that there is an even number of consecutive integers.This statement tells you nothing about the actual values of the integers, but the average of an even number of consecutive integers will never be an integer. Therefore, the average of the n consecutive integerscannot equal $1$.

(2) SUFFICIENT: You know that the sum of the n consecutive integers is positive, but smaller than n. Perhaps the most straight forward way to interpret this statement is to express it in terms of the average of the n numbers, rather than the sum. Using the formula $Average=Sum+Number$, you can reinter pret the statement by dividing the compound inequality by n:

$0<S<n\frac{0}{n}<\frac{S}{n}<\frac{n}{n}0<\frac{S}{n}<1$

This tells you that the average integer in set S is larger than $0$ but less than $1$. Therefore, the average number in the set does NOT equal $1$, so the statement is sufficient. The correct answer is (D).

As a footnote, this situation can happen ONLY when there is an even number of integers, and when the "middle numbers" in the set are $0$ and $1$. For example, the set of consecutive integers $\{0,1\}$ has a median number of $0.5$. Similarly, the set of consecutive integers $\{-3,-2,-1,0,1,2,3,4\}$ has a median number of $0.5$.