Suppose you have algorithms with the five running times listed below. (Assume these are the exact running times.) How much slower do each of these algorithms get when you (a) double the input size, or (b) increase the input size by one?
a) $n^2$ b) $n^3$ c) $100n^2$ d) $nlogn$ e) $2^n$
It is (b) $\frac {f (n+1)-f (n)}{f (n)} $and (a) $\frac {f (2n)-f (n)}{f (n)} $. For example, for $n^2$, if you increase the input size by one, the percentage increase is
$$\frac{(n+1)^2-n^2}{n^2}=\frac{2n+1}{n^2}\times 100\%$$
If you double the input size, the percentage increase is
$$\frac{(2n)^2-n^2}{n^2}=300\%$$