Challenge problems

[Analistul]

In the following answer to a challenge problem (pasting does not quite work and some of the data was not pasted) one of the sentences says:

"This can be generalized for any odd number n. That is, if there are an odd number n terms in a consecutive series of positive integers with first term k then (n-1)/2 = the middle term of the series."

I believe it should say "then THE FIRST TERM K PLUS (n-1)/2 = the middle term of the series", as the middle term of the series is, in reality, (n+1)/2.

Regards and thanks,



06/09/03
Question
Given a series of n consecutive positive integers, where n > 1, is the average value of this series an integer divisible by 3?

(1) n is odd

(2) The sum of the first number of the series and is an integer divisible by 3

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.



Answer
In order for the average of a consecutive series of n numbers to be an integer, n must be odd. (If n is even, the average of the series will be the average of the two middle numbers in the series, which will always be an odd multiple of 1/2.)

Statement (1) tells us that n is odd so we know that the average value of the series is an integer. However, we have no way of knowing whether this average is divisible by 3.

Statement (2) tells us that the first number of the series plus is an integer divisible by 3.

Since some integer plus yields another integer, we know that must itself be an integer.

In order for to be an integer, n must be odd. (Test this with real numbers for n to see why.)

Given that n is odd, let's examine some sample series:

If k is the first number in a series where n = 5, the series is { k, k + 1, k + 2, k + 3, k + 4 }. Note that . Thus, the first term in the series + = k + 2. Notice that k + 2 is the middle term of the series.

Now let’s try n = 7. The series is now {k, k + 1, k + 2, k + 3, k + 4, k + 5, k + 6}. Note again that . Thus, the first term in the series + = k + 3. Notice (again) that k + 3 is the middle term of the series.

This can be generalized for any odd number n. That is, if there are an odd number n terms in a consecutive series of positive integers with first term k then = the middle term of the series.

Recall that the middle term of a consecutive series of integers with an odd number of terms is also the average of that series (there are an equal number of terms equidistant from the middle term from both above and below in such a series, thereby canceling each other out). Hence, statement (2) is equivalent to saying that the middle term is an integer divisible by 3. Since the middle term in such a series IS the average value of the series, the average of the series is an integer divisible by 3.

Thus statement (2) alone is sufficient to answer the question and B is the correct answer choice.

[unique]

The cut pasting makes it unreadable. What is Statement 2 -cannot follow? Pls can u write it.

In the following answer to a challenge problem (pasting does not quite work and some of the data was not pasted) one of the sentences says:

"This can be generalized for any odd number n. That is, if there are an odd number n terms in a consecutive series of positive integers with first term k then (n-1)/2 = the middle term of the series."

I believe it should say "then THE FIRST TERM K PLUS (n-1)/2 = the middle term of the series", as the middle term of the series is, in reality, (n+1)/2.

Regards and thanks,



06/09/03
Question
Given a series of n consecutive positive integers, where n > 1, is the average value of this series an integer divisible by 3?

(1) n is odd

(2) The sum of the first number of the series and is an integer divisible by 3

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.



Answer
In order for the average of a consecutive series of n numbers to be an integer, n must be odd. (If n is even, the average of the series will be the average of the two middle numbers in the series, which will always be an odd multiple of 1/2.)

Statement (1) tells us that n is odd so we know that the average value of the series is an integer. However, we have no way of knowing whether this average is divisible by 3.

Statement (2) tells us that the first number of the series plus is an integer divisible by 3.

Since some integer plus yields another integer, we know that must itself be an integer.

In order for to be an integer, n must be odd. (Test this with real numbers for n to see why.)

Given that n is odd, let's examine some sample series:

If k is the first number in a series where n = 5, the series is { k, k + 1, k + 2, k + 3, k + 4 }. Note that . Thus, the first term in the series + = k + 2. Notice that k + 2 is the middle term of the series.

Now let’s try n = 7. The series is now {k, k + 1, k + 2, k + 3, k + 4, k + 5, k + 6}. Note again that . Thus, the first term in the series + = k + 3. Notice (again) that k + 3 is the middle term of the series.

This can be generalized for any odd number n. That is, if there are an odd number n terms in a consecutive series of positive integers with first term k then = the middle term of the series.

Recall that the middle term of a consecutive series of integers with an odd number of terms is also the average of that series (there are an equal number of terms equidistant from the middle term from both above and below in such a series, thereby canceling each other out). Hence, statement (2) is equivalent to saying that the middle term is an integer divisible by 3. Since the middle term in such a series IS the average value of the series, the average of the series is an integer divisible by 3.

Thus statement (2) alone is sufficient to answer the question and B is the correct answer choice.

[StaceyKoprince]

It looks like we had to use images to post really complicated math text, so I doubt he or she will be able to write it out cleanly. I'll have to go look this one up in the database so I can actually read it! Stay tuned.