2023 AMC 10B Problems/Problem 23
Solution 1
Since one of the terms was either 
more or less than it should have been, the sum should have been 
or 
The formula for an arithmetic series is 
This can quickly be rederived by noticing that the sequence goes 
, and grouping terms.
We know that 
or 
. Let us now show that is not possible.
If 
, we can simplify this to be 
Since every expression in here should be an integer, we know that either 
and 
or 
and 
The latter is not possible, since 
and 
The former is also impossible, as 
Thus, 
.
We can factor 
as 
. Using similar reasoning, we see that 
can not be paired as 
and , but rather must be paired as 
and 
with a factor of somewhere.
Let us first try 
Our equation simplifies to 
We know that 
so we try the smallest possible value: 
This would give us 
(Indeed, this is the only possible 
.)
There is nothing wrong with the values we have achieved, so it is reasonable to assume that this is the only valid solution (or all solutions sum to the same thing), so we answer 
For the sake of completeness, we can explore 
It turns out that we reach a contradiction in this case, so we are done.
~Technodoggo
Solution 2
There are 
terms, the 
th term is 
, summation is 
.
The summation of the set is 
. First, : its only possible factors are 
, and as said by the problem, 
, so must be 
or. Let's start with 
. Then, 
, and this means 
,
. Summing gives 
. We don't need to test any more cases, since the problem writes that all 
are the same, so we don't need to test other cases.
-HIA2020
