Multiply Each Time

The list 2, 10, 50, 250, ... multiplies by 5 each time. A list like this is a geometric sequence, and the number it multiplies by is its common ratio r. Here the first term is 2 and r is 5.

The 5th term is 4 multiplications after the first, because the first term has not been multiplied at all. So it is $2\cdot {5}^{4}$. Work the power first: ${5}^{4}$ is 625, and 2 times 625 is 1250. Multiplying 2 by 5 first would give ${10}^{4}$, a different number. In general, with a(1) the first term,

$$a\left(n\right) = a\left(1\right)\cdot {r}^{n-1}$$

Written as a function, the rule for 2, 10, 50, ... is $a\left(n\right) = 2\cdot {5}^{n-1}$.