If the terms of a sequence increase by a constant difference or constant ratio (these are both recursive descriptions), then the sequences are arithmetic or geometric, respectively, and we have closed formulas for each of these based on the initial terms and common difference or ratio.
If the terms of a sequence increase at a polynomial rate (that is, if the differences between terms form a sequence with a polynomial closed formula), then the sequence is itself given by a polynomial closed formula (of degree one more than the sequence of differences).
If the terms of a sequence increase at an exponential rate, then we expect the closed formula for the sequence to be exponential. These sequences often have relatively nice recursive formulas, and the characteristic root technique allows us to find the closed formula for these sequences.
If we want to prove that a statement is true for all values of \(n\) (greater than some first small value), and we can describe why the statement being true for \(n = k\) implies the statement is true for \(n = k+1\text{,}\) then the principle of mathematical induction gives us that the statement is true for all values of \(n\) (greater than the base case).