Question 1120323
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The first tutor found the solution with a positive constant ratio between terms; it can be found using a little educated trial and error and some mental arithmetic.<br>
Tutor @ikleyn showed there are two solutions.<br>
Here is a different approach to find both solutions.<br>
Since S2 = 20 and S3 = 65, we know T3 = 45.  So
{{{S2 = a+ar = 20}}}
{{{T3 = ar^2 = 45}}}<br>
Then<br>
{{{(a+ar)/(ar^2) = 20/45 = 4/9}}}
{{{(1+r)/r^2 = 4/9}}}
{{{4r^2 = 9r+9}}}
{{{4r^2-9r-9 = 0}}}
{{{(4r+3)(r-3) = 0}}}<br>
And we have what we need to find both solutions, one with r = -3/4 and one with r = 3.<br>
T3 = 45 and r = 3 gives us the first three terms as 5, 15, and 45;
T3 = 45 and r = -3/4 gives us the first three terms as 80, -60, and 45.