Question 1092202
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The third term is 24:<br>
{{{ar^2=24}}}
--> {{{a = 24/r^2}}}  (1)<br>
The sum of the first two terms is 90:<br>
{{{a+ar = a(1+r) = 90}}}  (2)<br>
Substitute (1) into (2):<br>
{{{(24/r^2)(1+r) = 90}}}
{{{24(1+r) = 90r^2}}}
{{{90r^2-24r-24 = 0}}}
{{{15r^2-4r-4 = 0}}}
{{{(5r+2)(3r-2) = 0}}}<br>
The two values of the common ratio are -2/5 and 2/3.<br>
For the common ratio -2/5, the first term a is<br>
{{{24/(-2/5)^2 = 24/(4/25) = 150}}}<br>
That gives us the sequence 150, -60, 24, ...<br>
For the common ratio 2/3, the first term a is<br>
{{{24/(2/3)^2 = 24/(4/9) = 54}}}<br>
That gives us the sequence 54, 36, 24, ...<br>