A finite sequence {An} is arithmetic. If each term An of the sequence

Question

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

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

NonPlus · Answer

IMO C

I solved it by plugging in sequences.
1. Take the sequence (4, 6, 8, 10), average is 7 (odd) and for sequence (4,8), average is (even). Hence, Insufficient.
2. Take the sequence (1, 2, 6) and (6, 8, 16). First will give 3(odd) and second will give 10(even). Insufficient

Combining both, we can check through sequences (2, 4, 6) (6, 8, 10) etc. we will get average as (even)

amanvermagmat · Answer

In an arithmetic sequence, arithmetic mean = median. So if number of terms is odd, then arithmetic mean = the middle term. But if number of terms is even, then arithmetic mean = average of the two middle terms.

(1) All terms are even numbers. 
so if number of terms of this sequence is odd, then median = the middle term, which would be an even number only. 
But if number of terms of this sequence is even, then median = average of the two middle terms = average of the two even numbers in the middle. Now average of two even numbers could be even (eg, that of 4 and 8) or it could be odd (eg, that of 4 and 6). So in this case arithmetic mean of entire sequence could be either odd or even. 
Not sufficient.

(2) Number of terms is odd. Clearly not sufficient.

But combining the second statement with first, since number of terms is odd, arithmetic mean of this sequence = median = the middle term, which is given to be an even number (statement 1). So its even. YES is our answer. Sufficient.

Hence  C answer

MathRevolution · Answer

=>

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 A k  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  \frac{(2 + 4)}{2} = 3 , which is an odd number. The answer is “no”.

If the sequence is {2,4,6}, then the average is  \frac{(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.

poorvi125 · Answer

Consider this, combining both the situations:

There are three terms (i.e., odd no. of terms) - 2, 14, 10. Also notice that all these are even (so condition is satisfied). Average of these is - 26/3 = which isn;t even an integer, let alone an even number.

IMO, the answer is E.

ThatDudeKnows 
 Bunuel

can you please help?

ThatDudeKnows · Answer

poorvi125

The definition of an arithmetic sequence is that you add a fixed amount (it can be positive or negative) between each two consecutive terms, so (2, 14, 10) doesn't fit the parameters. Armed with that definition, are you able to reason through the solution?

poorvi125 · Answer

Yes!
 ThatDudeKnows

Thankyou for clearing my doubt.

dhruvshawarma · Answer

Nice catch - i did not assume this to be an arithmetic progression !!!

A finite sequence {An} is arithmetic. If each term An of the sequence

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A finite sequence {An} is arithmetic. If each term An of the sequence

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