If S is a set of n consecutive integers, what is the sum of the intege

Question: Sum of integers in S = ?

Statement 1: n = 11

We do not know which 11 consecutive terms are in set S
Terms may be from 1 to 11 or from 2 to 12 hence

NOT SUFFICIENT

Statement 2:The sum of the least integer and the greatest integer in S is zero.

i.e. SUm of all terms will be zero for any integer value of n



SUFFICIENT
ANswer: Option B

IMO B

Statement 1: n=11
Now the set no.s Could be from 1 to 11, -7 to 5 and so on.. (insufficient)

Statement 2: let first no. Be A-n and last no. Be a+n (it has be a set of odd no.)

A+n+A-n = 0
2A= 0
A=0 (median is zero)
Now n could be 3,5,7,.... So on. We are asked for a specific value. What's the sum of integers of set S.
The sum will always be zero. Since the upper half of median will cancel out with lower half.
Statement 2 is sufficient.
Answer is B

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maths revolution team,

Can you please solve the problem with Variable approach method ??

A detailed explanation will help.

Average = (Sum)/number of terms in the set.
this is a set of consecutive integers. therefore average is (First term + Last term)/2.
(First term + Last term)/2 = (Sum)/Number of terms in the set.

(1) n = 11

statement1 alone is not sufficient.

(2) The sum of the least integer and the greatest integer in S is zero.

Since, (First term + Last term) = 0
0/2 = (Sum)/Number of terms in the set => sum = 0. Sufficient.
Answer - B

Statement I: # of terms in S =11. It can be {1,2,3,..11} or {5,6,7 .....15}. INSUFFICIENT. Options A & D gone.

Statement II: the sum of the least integer and the greatest integer in S is zero i.e. it has both positive and negative numbers. Since sum of least + Greatest = 0 ==> set is of form:
{-1,0,1}. or {-2,-1,0,1,2}... Set can be anything but for every +ve number it has corrresponding -ve number. Sum will always be zero.
Definite answer. Sufficient.

Hence B.
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Inference from the question stem:
S is a set of n consecutive Integers => the increment between any two consecutive numbers is Fixed.
All such sets where the difference between any two consecutive numbers is the same are defined as Evenly Spaced Sets.
So, above set S is an Evenly spaced set

Few facts about the Evenly spaced sets:
[1) Mean of the set is always equal to the median of the set
2) Average of the first and last terms in the set is equal to mean of the set

Sum = Mean * number of terms in the given set
Using the facts, along with the above formula we can very easily find the sum of all numbers in the given evenly spaced set!

Now Analyze Stat (1) alone:
1) n = 11
we only know the number of terms in the set but we need mean of the set also to find the Sum of integers in the set

Thus, stat (1) alone is not sufficient to find the answer to the given question



Analyze Stat (2) alone:
2) The sum of the least integer and the greatest integer in S is zero.

We know that mean of the evenly spaced set is same as the average of the first and last terms in the set
here, as the sum of the least integer( First Term) and the greatest integer(Last term) in S is zero, thus the Mean of the first and last terms in the set is also Zero
which implies the sum of the set is also Zero as
Sum = Mean * number of terms in the given set

Thus, stat (2) alone is sufficient to find the sum of integers in the set S
Hence, the answer is B

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