# How do you find the explicit formula for: 1, 2, 4, 8...?

Feb 18, 2016

${a}_{n} = {2}^{n - 1}$

#### Explanation:

The difference between successive terms is not a constant,
so the sequence is not an arithmetic sequence.

Checking to see if the sequence might be geometric,
we see the successive terms are all a multiple of $2$ times the previous term.

${a}_{1} = 1 = {2}^{0}$
${a}_{2} = {a}_{1} \times 2 = {2}^{1}$
${a}_{3} = {a}_{2} \times 2 = {a}_{1} \times {2}^{2} = {2}^{2}$
${a}_{4} = {a}_{3} \times 2 = {a}_{1} \times {2}^{3} = {2}^{3}$

so we could generalize:
$\textcolor{w h i t e}{\text{XXX}} {a}_{n} = {2}^{n - 1}$