Data Sufficiency

[GMAT math practice question]

A finite sequence {An} is arithmetic. If each term An of the sequence is an integer, is the average (arithmetic mean) of the terms of {An} an even number?

1) All of the terms are even numbers.
2) The number of terms is an odd number

Max@Math Revolution wrote:[GMAT math practice question]

A finite sequence {An} is arithmetic. If each term An of the sequence is an integer, is the average (arithmetic mean) of the terms of {An} an even number?

1) All of the terms are even numbers.
2) The number of terms is an odd number

An arithmetic sequence is an EVENLY SPACED SET.
In an evenly spaced set, AVERAGE = MEDIAN.
Question stem, rephrased:
Is the median of the set even?

Statement 1:
Case 1: {0, 2}
In this case, the median of the set is 1, so the answer to the rephrased question stem is NO.
Case 2: {0, 2, 4}
In this case, the median of the set is 2, so the answer to the rephrased question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statement 2:
Case 1: {0, 1, 2}
In this case, the median of the set is 1, so the answer to the rephrased question stem is NO.
Case 2: {0, 2, 4}
In this case, the median of the set is 2, so the answer to the rephrased question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statements combined:
Since the set contains an odd number of terms, the median is equal to the middle value.
Since all of the values are even, the middle value -- and thus the median -- must be even.
Thus, answer to the rephrased question stem is YES.
SUFFICIENT.

The correct answer is C.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

A finite arithmetic sequence can be specified using the first term, the common difference and the number of terms. This means that three variables are required to specify and arithmetic sequence, and so E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.


Conditions 1) & 2):
Since n is an odd number, we have n = 2k + 1 for some integer k.
Since the sequence is arithmetic, the median Ak is the average of the sequence.
Since all of the terms are even numbers, Ak is an even number.
Both conditions are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4 (A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If the sequence is {2,4}, then the average is (2 + 4)/2 = 3, which is an odd number. The answer is "no".

If the sequence is {2,4,6}, then the average is (2 + 4 + 6)/3 = 4, which is an even number. The answer is "yes".
By CMT 2, condition 1) is not sufficient since it does not lead to a unique answer.

Condition 2)
If the sequence is {1, 2, 3}, then the average is 2, which is an even number. The answer is "yes".
If the sequence is {1, 3, 5}, then the average is 3, which is an odd number. The answer is "no".
By CMT 2, condition 2) is not sufficient since it does not lead to a unique answer.

Therefore, C is the answer.

Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.