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

Apr 4, 2016

${a}_{n} = {2}^{\left(n + 1\right)}$

#### Explanation:

Let any term in the sequence by $a$
Then the ${n}^{t h}$ term is ${a}_{n}$

First write this out:

color(white)("d")

$n \textcolor{w h i t e}{.} \to \textcolor{w h i t e}{\text{d") 1color(white)("d") 2color(white)("d") 3color(white)("dd") 4color(white)("dd}} 5$
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 ${a}_{n} = 2 \times {a}_{n - 1}$

So, for example, term 4 is $2 \times \text{ term 3}$

$n \textcolor{w h i t e}{.} \to \textcolor{w h i t e}{\text{d")1color(white)("d.d") 2 color(white)("ddd") 3 color(white)("d..") 4 color(white)(d"d}} 5$
$\underline{{a}_{n} \textcolor{w h i t e}{\text{d")->4color(white)("ddd") 8color(white)("dd") 16 color(white)("d.")32color(white)("d,}} 64}$
$\textcolor{w h i t e}{\text{dddddd}} {2}^{2} \textcolor{w h i t e}{. .} {2}^{3} \textcolor{w h i t e}{. .} {2}^{4} \textcolor{w h i t e}{. .} {2}^{5} \textcolor{w h i t e}{. .} {2}^{6}$

But we need to relate the $x$ in ${2}^{x}$ to $n$

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

So we have ${a}_{n} = {2}^{\left(n + 1\right)}$