For what positive value of p will this be a geometric sequence: p-3, p+1, 3p+3?

Jan 16, 2016

$p = 5$

Explanation:

If $p - 3 , p + 1 , 3 p + 3$ is a geometric sequence
then for some constant $c$:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\left(p - 3\right) \times c = \left(p + 1\right)}$
and
color(white)("XXX"color(blue)((p+1)xxc=3p+3)

Therefore
$\textcolor{w h i t e}{\text{XXX}} \frac{\textcolor{b l u e}{\left(\left(p + 1\right) \times \cancel{c}\right)}}{\textcolor{red}{\left(\left(p - 3\right) \times \cancel{c}\right)}} = \frac{\textcolor{b l u e}{\left(3 p + 3\right)}}{\textcolor{red}{\left(p + 1\right)}} = \frac{3 \left(\cancel{p + 1}\right)}{\cancel{p + 1}}$

$\textcolor{w h i t e}{\text{XXX}} \frac{p + 1}{p - 3} = 3$

$\textcolor{w h i t e}{\text{XXX}} p + 1 = 3 p - 9$

$\textcolor{w h i t e}{\text{XXX}} - 2 p = - 10$

$\textcolor{w h i t e}{\text{XXX}} p = 5$