n integers consecutive

[jain2016]

A list contains n distinct integers. Are all n integers consecutive?

(1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.

(2) The positive difference between any two numbers in the list is always less than n.


OAD

Hi Experts ,

Please explain statement 1.

Many thanks in advance.

SJ

[chetan.sharma]

A list contains n distinct integers. Are all n integers consecutive?

(1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.

(2) The positive difference between any two numbers in the list is always less than n.


OAD

Hi Experts ,

Please explain statement 1.

Many thanks in advance.

SJ

HI,
since you have asked for Statementr I, I will restrict myself to statement 1 only..
(1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.
MOST IMPORTANT here is to know what is the average of consecutive numbers..
it is the middle number if number of integers are ODD and average of two middle number if EVEN number integers are there..

lets put this in the statement above..
1) if there were odd numbers, average is middle number..
I remove the smallest number.. number s become even and the average becomes 0.5 less than the previous average.
I remove the largest, the average shifts 0.5 above the previous one. basically center of new MIDDLE two.
OVERALL difference = 0.5-(-0.5)=1..
example ..
1,2,3,4,5 ..
3 is average..
remove 1, average becomes center of 2 and 3= 2.5..
remove 5, average becomes center of 3 and 4 = 3.5
Difference = 1
You can try with EVEN number of integers too

so SUFF
hope it helps