Question

How do you write the explicit formula for the sequence 4,8,16,32,64...?

Answer 1

`a_n = 2^((n+1))`

Explanation:

Let any term in the sequence by `a`
Then the `n^(th)` term is `a_n`

First write this out:

color(white)("d")

`ncolor(white)(.) -> color(white)("d") 1color(white)("d") 2color(white)("d") 3color(white)("dd") 4color(white)("dd") 5`
`a_n->color(white)("d")4color(white)("d") 8 color(white)("d") 16color(white)("d") 32 color(white)("d") 64"`

Apart from the starting point notice that each term is `a_n = 2xx a_(n-1)`

So, for example, term 4 is `2xx" term 3"`

`ncolor(white)(.) -> color(white)("d")1color(white)("d.d") 2 color(white)("ddd") 3 color(white)("d..") 4 color(white)(d"d") 5`
`underline(a_ncolor(white)("d")->4color(white)("ddd") 8color(white)("dd") 16 color(white)("d.")32color(white)("d,") 64)`
`color(white)("dddddd")2^2color(white)(..)2^3color(white)(..)2^4color(white)(..)2^5color(white)(..)2^6`

But we need to relate the `x` in `2^x` to `n`

Notice that in each case `x=n+1`

So we have `a_n = 2^((n+1))`