Question 1141053
For what values of x will (...,x+1, x+5, 3x+2, ...) be a geometric sequence?
<pre>It will be a geometric sequence with common ratio r if but only if,
BOTH of the following are true:

1. the second term x+5 is r times the first term x+1, that is,

      x+5 = r(x+1)

AND

2. the third term 3x+2 is r times the second term x+5, that is,

     3x+2 = r(x+5)

So we have the system of two equations in two unknowns x and r:

{{{system(x+5 = r(x+1), 3x+2 = r(x+5))}}}

Solve the first equation for r:

{{{r=(x+5)/(x+1)}}}

Substitute that for r in the second equation:

{{{3x+2 = r(x+5))}}}
{{{3x+2 = ((x+5)/(x+1))(x+5))}}}

Multiply both sides by (x+1)

{{{(3x+2)*red((x+1)) = ((x+5)/(x+1))(x+5)*red((x+1))}}}

{{{3x^2+5x+2=((x+5)/(cross(x+1)))(x+5)*red((cross(x+1)))}}}

{{{3x^2+5x+2=(x+5)(x+5)}}}

{{{3x^2+5x+2=x^2+10x+25}}}

{{{2x^2-5x-23=0}}}

Oh darn! That doesn't factor, so we must use the quadratic formula
(Are you sure you copied it right?)

{{{x = (-b +- sqrt( b^2-4ac ))/(2a) }}}

with a=2, b=-5, c=-23

{{{x = (-(-5) +- sqrt( (-5)^2-4(2)(-23) ))/(2(2)) }}}

{{{x = (5 +- sqrt( 25-(8)(-23) ))/4 }}}

{{{x = (5 +- sqrt( 25+184 ))/4 }}}

{{{x = (5 +- sqrt(209))/4 }}}

Edwin</pre>