# How can I solve this inequation over positive integers?

$$2023^i + 2023^{101-i}(-1)^i < 0, \quad i\in \mathbb N^+$$

My code:

Reduce[2023^x+2023^(101-x)*(-1)^x<0 && x ∈ PositiveIntegers, x]


gives an error message :

Reduce::nsmet: This system cannot be solved with the methods available to Reduce.

How can I solve this problem correctly?

• This can be solved even without pen and paper just looking at it. Commented Apr 28 at 13:53

Help it a little

Reduce[2023^x + 2023^(101 - x)*(-1)^x < 0 &&  x < 100 && x ∈ PositiveIntegers, x]


gives

x == 1 || x == 3 || x == 5 || x == 7 || x == 9 || x == 11 || x == 13 ||
x == 15 || x == 17 || x == 19 || x == 21 || x == 23 || x == 25 ||
x == 27 || x == 29 || x == 31 || x == 33 || x == 35 || x == 37 ||
x == 39 || x == 41 || x == 43 || x == 45 || x == 47 || x == 49


To verify you could do this

e = 2023^x + 2023^(101 - x)*(-1)^x;
sol = Reduce[e < 0 && x < 100 && x ∈ PositiveIntegers, x];
values = Last[#] & /@ List @@ sol;
{#, N[e /. x -> #]} & /@ values;
TableForm[%, TableHeadings -> {None, {"x", "expression"}}]