The first $n$ consecutive positive integers in a series are $1, 2, 3, 4...n-1, n$. The integers $a, b, c, d$ are removed from the list. $100<a<b<c<d$ and at least $3$ of $a, b, c, d$ are consecutive the average of the remaining integers is $89.5625$ and i need help finding all the possible values of $d$.
$86.5625 = 89 \dot 9/16$. I made the sum of $a+b+c+d = s$. And I know that the average of the numbers before removed is $89\cdot 9/16 + 0.5\cdot2 \approx 91 9/16$. And since the average is equal to around the middle number then so $n$ is around $183$. After this point I don't know what to do.
After removing, you have $n-4$ different positive integers. Their average is $89.5625$, and since this is at least the average of the first $n-4$ positive integers, you must have $n-4<179$.
Also, the total of these numbers is $(n-4)\times 89.5625$. For this to be an integer, $n-4$ must be a multiple of $16$. Also, you know $(n-4)\times 89.5625$ is less than the sum of the first $n$ positive integers. These two facts should allow you to determine the value of $n$ (only one $n$ is possible).