Question 118821
Find the value of x if the first three terms of a geometric sequence are given as:
{5x-8, 3x, 2x+2,...}
You know that in a geometric sequence, consecutive terms have a common ratio.
This can be expressed as:
{{{a[n]/a[n-1]  = k}}} or, in words...Any term divided by the previous term is a constant called the common ratio, r.
So, let's use this useful fact to find x in your problem:
{{{3x/(5x-8) = k}}} and...
{{{(2x+2)/3x = k}}} and, since k = k, we can set these two equal to each other:
{{{3x/(5x-8) = (2x+2)/3x}}} Simplify and solve for x.
{{{3x(3x) = (5x-8)(2x+2)}}} Expand.
{{{9x^2 = 10x^2-6x-16}}} Subtract {{{9x^2}}} from both sides.
{{{0 = x^2-6x-16}}}Solve this quadratic equation by factoring.
{{{(x+2)(x-8) = 0}}} Apply the zero product principle:
{{{x+2 = 0}}} or {{{x-8 = 0}}}
If {{{x+2 = 0}}} then {{{x = -2}}} or
if {{{x-8 = 0}}} then {{{x = 8}}}
So there are two possible solutions for x.
{{{x = -2}}}or {{{x = 8}}} and since the problem does not give us the common ratio, both values of x are valid solutions.
Check:
{5x-8, 3x, 2x+2,...} Let's try x = -2
{5(-2)-8, 3(-2), 2(-2)+2,...}
{-18, -6, -2,...} The common ratio, r, is:
{{{-6/-18 = 1/3}}}
{{{-2/-6 = 1/3}}} OK
Now try x = 8:
{5x-8, 3x, 2x+2,...}
{5(8)-8, 3(8), 2(8)+2,...}
{32, 24, 18,...} The common ratio here is:
{{{24/32 = 3/4}}}
{{{18/24 = 3/4}}} OK