Suggest inductive definitions which would produce the following sequences:
1) $\displaystyle 2, 4, 6, 8, 10 ...$
2) $\displaystyle \frac{1}{3},\frac{1}{8}, \frac{1}{27}, \frac{1}{81}$
Whats the formula to produce the definition?
Well, let's just look at them individually:
1) $\displaystyle 2, 4, 6, 8, 10, ...$
It's simple to find what produced this sequence, just take a look at it:
$\displaystyle (2*1), (2*2), (2*3), (2*4), (2*5), ...$
It looks like the sequence is defined by:
$\displaystyle \color{blue}a_n = 2n$
2) $\displaystyle \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, ...$
Let's take a closer look:
$\displaystyle \frac{1}{3^1}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, ...$
It looks like this one is defined by:
$\displaystyle \color{red}a_n = \frac{1}{3^n}$
And there you go.