text: detrmine by induction all the positive integers valuesof n for which n^3>2^n .prove your claim by mathematica   for n=0 the statement is false ("0>1") for n=1 the statement is false too ("1>2") yet, for n=2 it is true: (8>4)  ...  if you do an empyrical investigation, you discover that the inequality is true for all n between 2 and 9 (including the extremes)  how can you prove this by induction?  write the inequality for n+1: (n+1)^3 > 2^(n+1) develop the cube and the power: n^3 + (3n^2+3n+1) > 2*2^n (=2^n + 2^n) your inequality is true until 3n^2+3n+1 > 2^n draw exactly the parabola and the exponential and get your interval of truth.  i hope that was what you needed.  p.s. try it with all n >= 10 : the statement will prove itself to be false