1. ## Sequence Forumla?

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?

2. 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.

3. That was easy.. how can I solve these:
$\displaystyle 1) b-2c, b-c, b+c, ...$
$\displaystyle 2) 1, 1+x, (1+x)^2, (1+x)^3, ...$

4. 1)

$\displaystyle \sum_{k=-2}^{n}b+kc$

or

$\displaystyle a_n = b + nc$

2)

$\displaystyle a_n = (1 + x)^n$

or

$\displaystyle \sum_{k=0}^{n}(1 + x)^k$