+618 The set $\{2, 4, 6, \dots, n\}$ contains the positive consecutive even integers from 2 through $n$. When one of the integers from the set is removed, the average of the remaining integers in the set is 28. What is the least possible value of $n$ ?
Since the set consists of even integers from: 2, 4, 6, 8.........n, and if the average of the set is 28, then 28th term must be=28 x 2 =56=n. The sum of 28 terms from 2+4+ 6+ 8.......56=812. 812/28 =29 average. Therefore: 812 - 56 =756 / 27=28 - the average.
Therefore, the least possible value of n = 56.
+128406 I get a slightly different answer than the guest does
Notice that the greatest possible average from adding the digits 2 + 4 + 6 + ....+ n
is given by :
[ ( n / 2) (n / 2 + 1 ) - 2 ] / [ (n - 2) / 2 ]
Example.... 2 , 4 , 6 , 8 10
And the greatest average possible average is produced when the least number, 2, is omitted
And this average is [ ( 10 / 2) (10 / 2 + 1 ) - 2 ] / [ (10 - 2) / 2 ] =
[(5)(6) - 2] / (8 / 2 ) = 28 / 4 = 7
So...we want to solve this :
[ ( n / 2) (n / 2 + 1 ) - 2 ] / [ (n - 2) / 2 ] = 28 simplify
2 [ n^2 / 4 + n/2 - 2 ] = 28 [ n - 2 ]
2 [ ( n^2 + 2n - 8) / 4 ] = 28 [n - 2]
( n^2 + 2n - 8) / 2 = 28 [n - 2]
( n^2 + 2n - 8) = 56 [n - 2]
n^2 + 2n - 8 = 56n - 112
n^2 - 54n + 104 = 0 factor
( n - 52) ( n - 2) = 0
And its obvious that n = 52 and this is the smallest value of n
Proof :
2 + 4 + 6 + ....... + 52 can be written as
1 + 2 + 3 + ........ + 26
+
1 + 2 + 3 + ......... + 26
So 52 is the 26th term
And the sum of these two series = 2 (26)(27)/ 2 = 26 * 27
And the average of this series omitting the first term is given by
[Series sum - 2] / [ Number of remaining terms omitting the first one, 2 ]
[ 26 * 27 - 2 ] / 25 =
[ 702 - 2] / 25 =
700 / 25 =
28
