Suppose n is an integer ≥2. Determine the first digit after the decimal point in the decimal expansion of the number (n3+2n2+n)1/3.
Testing values of n implies the answer is the digit 6.
If that is true then the compound inequality must be true:
(n+0.6)^3 < n^3+2n^2+n < (n+0.7)^3
Each half of the compound inequality can be reduced to
0 < n^2+0.4n-1.08 and 0 < n^2+0.47n+3.43
The first quadratic is true for n>1.2583 and the second is true for n>-0.903. So combining those and applying integer domain restrictions then we get n>=2. QED
Solution
