If the average (arithmetic mean) of n consecutive odd integers is 10,

If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

Odd consecutive integers is an evenly spaced set. For any evenly spaced set the mean equals to the average of the first and the last terms, so in our case \(mean=10=\frac{x_1+x_{n}}{2}\) --> \(x_1+x_{n}=20\). Question: \(x_1=?\)

(1) The range of the n integers is 14 --> the range of a set is the difference between the largest and smallest elements of a set, so \(x_{n}-x_1=14\) --> solving for \(x_1\) --> \(x_1=3\). Sufficient.

(2) The greatest of the n integers is 17 --> \(x_n=17\) --> \(x_1+17=20\) --> \(x_1=3\). Sufficient.

Answer: D.
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If mean of consecutive odd integers is 10, the sequence of numbers will be something like this:
9, 11 or
7, 9, 11, 13 or
5, 7, 9, 11, 13, 15 or
3, 5, 7, 9, 11, 13, 15, 17 or
1, 3, 5, 7, 9, 11, 13, 15, 17, 19
etc
Every time you add a number to the left, you need to add one to the right to keep the mean 10. The smallest sequence will have 2 numbers 9 and 11, the largest will have infinite numbers.

Stmnt 1: Only one possible sequence: 3, 5, 7, 9, 11, 13, 15, 17 will have range 14. Least of the integers is 3. Sufficient.
Stmnt 2: Only one possible sequence:3, 5, 7, 9, 11, 13, 15, 17
Least of the integers is 3. Sufficient.
Answer (D).

Note: You don't actually have to do all this. All such sequences will have distinct number of elements, greatest number, smallest number and range. So each statement alone will be sufficient.
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What we know:
If n consecutive integers have arithmetic mean of 10; those numbers are distributed either side of 10 perfectly symmetrically.

Statement (1)

Range is the difference between the greatest and the least numbers in the series.
If the range is known the least of these integers must be less than the mean, i.e. 10 - (14/2) = 3 (Note we don't really need to calculate this... to know we can is sufficient.)

Statement (2)

Similarly, the distance from the greatest number to mean equals to the distance from mean to the least number. So knowing the greatest number also sufficient to calculate the least number. (in this case 10 - (17 - 10) = 3)


The answer is D : Both Statement 1 and Statement 2 are sufficient ALONE .

Bunuel ,

My question is from the problem statement itself only one solution is possible. [ 3,5,7,9,11,13,15,17]. Are there any chances to encounter such a question on actual exam

Unfiniftley many sets are possible:
{9, 11}
{7, 9, 11, 13}
{5, 7, 9, 11, 13, 15}
...
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Hi Bunuel

Could you please explain what evenly spaced set means?

Evenly spaced set (aka arithmetic progression) is a special type of sequence in which the difference between successive terms is constant. Fore example, 1, 4, 7, 10, ... is an evenly spaced set. Check for more here: https://gmatclub.com/forum/math-sequenc ... 01891.html

Hope it helps.
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(2) The greatest of the n integers is 17
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Whoa, what has happened to me. I read 14 as 4 unfortunately

Can someone please explain me why we haven't taken negative odd integers into consideration ?

Bunuel why did you assume from the beginning that there are two odd consecutive integers ? and not more

I think you missed the highlighted part. x1 is the first term of the sequence and xn is the nth term of the sequence.
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For any evenly spaced set:
average = median




Since the average of the consecutive odd integers = 10, the median of the set must also be 10.

Statement 1:
For a range of 14 to be yielded:
The smallest integer in the set must be 7 places to the left of 10 --> 10-7 = 3
The greatest integer in the set must be 7 places to the right of 10 --> 10+7 = 17
Thus, the smallest integer = 3
SUFFICIENT.

Statement 2:
Since the greatest integer is 7 places to the right of 10, the smallest integer must be 7 places to the left of 10:
10-7 = 3
SUFFICIENT.


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If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

The average of an evenly spaced set must be the median. Since 10 is even, we must have an even number of integers in the set.

9 + 11 = average 10
3 + 5 + 7 + 9 + 11 + 13 + 15 + 17= average 10

To determine the least of the integers, we need to know the number of integers, the greatest number in the set, or the range.

(1) The range of the n integers is 14

SUFFICIENT.

(2) The greatest of the n integers is 17

SUFFICIENT.

Answer is D.
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If the average is 10, then 2 middle numbers are 9 and 11
(1) Range of the sequence is 14. The smallest and the largest number of the sequence is symmetric though 10.
Largest - Smallest = 14 --> (10+a) - (10-a) = 14 --> a = 7
--> Smallest number is 3, largest is 17, then we can count how many number
--> Suff

(2) We know the largest number --> Suff (similar to (1))
--> Answer D

Hi KarishmaB

With only D in our hand we can also assume the following; -1,1,3,5,7,9,11,13,15,17
why didn't we consider negative numbers and instead cut it at 3?

Because the average of this list is 8, not 10.

The average will be the middle term. So here average will be middle of 7 and 9.

When you added 1 and -1 to the list, the sum did not change but the average did because number of terms increased.
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Question Stem Analysis:

We are told that the average of n consecutive odd integers is 10. Remember that in an evenly spaced set, mean = median, so the median of this set is also 10. Since the median is even, n must also be even, so the median is the average of the two numbers in the middle. Thus, the possibilities for the set are:

{9, 11}

{7, 9, 11, 13}

{5, 7, 9, 11, 13, 15}

and so on.

Statement One Alone:

\(\Rightarrow\) The range of the n integers is 14

Let's calculate the ranges for the pattern we obtained above:

{9, 11}, range = 2

{7, 9, 11, 13}, range = 6

{5, 7, 9, 11, 13, 15}, range = 10

We see that the range increases by 4 each time. Following the pattern, we can determine the set with a range of 14 and determine the smallest integer in that set. Actually, the next set in our pattern is the set we are looking for: {3, 5, 7, 9, 11, 13, 15, 17}. Statement one alone is sufficient.

Eliminate answer choices B, C, and E.

Statement Two Alone:

\(\Rightarrow\) The greatest of the n integers is 17

Following the same process above, we can keep listing the possibilities until we hit the set where the largest element is 17. Once we determine the set, we can determine its smallest element. Statement two alone is sufficient.

Answer: D
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