Suppose [tex]a_n[/tex] is the number of possible combinations for a suitcase with a lock consisting of [tex]n[/tex] wheels. If you added one more wheel onto the lock, there would only be 9 allowed possible digits you can use for the new wheel. This means the number of possible combinations for [tex]n+1[/tex] wheels, or [tex]a_{n+1}[/tex] is given recursively by the formula
[tex]a_{n+1}=9a_n[/tex]
starting with [tex]a_1=10[/tex] (because you can start the combination with any one of the ten available digits 0 through 9).
For example, if the combination for a 3-wheel lock is 282, then a 4-wheel lock can be any one of 2820, 2821, 2823, ..., 2829 (nine possibilities depending on the second-to-last digit).
By substitution, you have
[tex]a_{n+1}=9a_n=9^2a_{n-1}=9^3a_{n-2}=\cdots=9^na_1=10\times9^n[/tex]
This means a lock with 55 wheels will have
[tex]a_{55}=10\times9^{54}[/tex]
possible combinations (a number with 53 digits).
