An arithmetic progression is one in which each subsequent term is the

Question

An arithmetic progression is one in which each subsequent term is the sum of the preceding number and a constant. If a, b, c, d, e, and f are integers in arithmetic progression, what can be said about the six terms for sure?

I. The average is not an integer.
II. The median is not an integer.
III. The median and the mean are equal.

A. II only
B. III only
C. I and III only
D. II and III only
E. I, II, and III

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rosyln · Answer

IMHO  B is the correct answer.
1. The average is not an integer: The average is calculated as (a + b + c + d + e + f)/6. Since a, b, c, d, e, f are integers, their sum sum is an integer. Dividing by 6, the average may or may not be an integer, depending on the sum. So this statement cannot be concluded for sure. 
2. The median is not an integer: The median is the average of the 3rd and 4th terms. As these are integers, the median also, may or may not be an integer. This statement cannot be concluded for sure.
3. The median and the mean are equal: In any AP, the mean and median are always equal. This statement is true. Hence, the answer is B) III only.

prantorboni · Answer

An arithmetic progression is one in which each subsequent term is the sum of the preceding number and a constant. If a, b, c, d, e, and f are integers in arithmetic progression, what can be said about the six terms for sure?

I. The average is not an integer. Since, we don't know anything about value of a, b, c, d, e, and f, we cann't draw conclusion based on these information. 
II. The median is not an integer. Since, we don't know anything about value of a, b, c, d, e, and f, we cann't draw conclusion based on these information. 
III. The median and the mean are equal.

Let's Assume, a=1 & the constant is 1, so then the sequence will be,
1, 2, 3, 4, 5, 6 where mean and median is same,
of if we assume a=1 & constant=2, sequence will be,
1, 3, 5, 7, 9, 11 where mean and median is same.

So, Answer: B. III only.

AbhinayaGMAT30 · Answer

Since the sequence is an arithmetic progression of 6 numbers, it can be written as follows:-

a, a+k, a+2k, a+3k, a+4k, a+5k

where, a is the first number of the arithmetic progression and k is the constant. Here, a is an integer as given in the question and k has to be an integer (otherwise the second term of the sequence would not be an integer).

Now,  mean  = (sum of above 6 numbers)/6 = (6a+15k)/6 = a+2.5k
 median  = (third term + fourth term)/2 = (2a+5k)/2 = a+2.5k

Hence, the third statement is definitely true.

Now, consider the Mean (or Median)= a+2.5k, where both a and k are integers.

Either k would be an even integer or an odd integer.

If K is an even integer, the Mean (or Median) = integer

If K is an odd integer, the Mean (or Median) = not an integer

Hence, the first and second statement in the questions can not be definitely true.

OPTION (B) is the answer then.

An arithmetic progression is one in which each subsequent term is the

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An arithmetic progression is one in which each subsequent term is the

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