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Both functions and is increasing if n is positive integer, which means their sum is increasing too. 5^3=125, so, the easiest way to prove the statement is to calculate for integers from 1 to 4.
If n is negative integer, then n^2 + n^3 ≤ 0, so it cannot be equal to 100.
If n = 0 then f(0) = 0.
The statement has been proven.