Quick questions on Growth and Decay - TCE Mathematics Applications (Tasmania)

A recurrence relation defines each term from the previous one and a starting value:

A frequent task asks not for a value but for the time to reach a target: how long for an investment to double, or for a drug to fall below a threshold. Set the closed form equal to the target and solve for $n$. Because $n$ is in the exponent, this needs logarithms or a guess-and-check on the calculator. For example, to find when $t_0 R^n$ first exceeds a target $T$, solve $R^n = \frac{T}{t_0}$, then $n = \frac{\log(T / t_0)}{\log R}$, and round up to the next whole period because the target is reached part way through.