Let <em>x</em> be the first term in the geometric sequence. Then the next two terms in the sequence are <em>xr</em> and <em>xr</em> ², where <em>r</em> is some constant. (This is the defining characteristic of geometric sequences.)
The sum of the first three terms is 42, so
<em>x</em> + <em>xr</em> + <em>xr</em> ² = 42
<em>x</em> (1 + <em>r</em> + <em>r</em> ²) = 42
The third term is 3.2 times the sum of the other two, so that
<em>xr</em> ² = 3.2 (<em>x</em> + <em>xr</em> )
Solve the second equation for <em>r</em> :
<em>xr</em> ² = 3.2 <em>x</em> (1 + <em>r</em> )
We can divide both sides by <em>x</em> since <em>x</em> ≠ 0. (This is obvious, since if <em>x</em> <u>was</u> zero, then all three terms in the sequence would be 0.)
<em>r</em> ² = 3.2 (1 + <em>r</em> )
<em>r</em> ² = 3.2 + 3.2<em>r</em>
<em>r</em> ² - 3.2<em>r</em> - 3.2 = 0
<em>r</em> ² - 16/5 <em>r</em> - 16/5 = 0
5<em>r</em> ² - 16<em>r</em> - 16 = 0
(5<em>r</em> + 4) (<em>r</em> - 4) = 0
==> <em>r</em> = -4/5 or <em>r</em> = 4
Since there are two possible values of <em>r</em> that might work, there are two possible sequences that meet the criteria.
Plug either of these solutions into the first equation:
<em>r</em> = -4/5 ==> <em>x</em> (1 + (-4/5) + (-4/5)²) = 42
… … … … … … 21/25 <em>x</em> = 42
… … … … … … <em>x</em> = 50
<em>r</em> = 4 ==> <em>x</em> (1 + 4 + 4²) = 42
… … … … … 21<em>x</em> = 42
… … … … … <em>x</em> = 2
Then the two possible answers would be
• if <em>r</em> = -4/5, then the three terms are {50, -40, 32}
• if <em>r</em> = 4, then they are {2, 8, 32}