Alg8.19

The sum of the first three terms of a GP is $\frac{39}{10}\,$ and their product is $1\,$ .Find the terms.

Let the three terms be $\frac{a}{r},a,ar\,$

Then given their product is $\frac{a}{r}\cdot a\cdot ar=1,a=1\,$

Then $\frac{1}{r}+1+r=\frac{39}{10},\frac{1+r+r^2}{r}=\frac{39}{10}\,$

$10(1+r+r^2)=39r,10r^2-19r+10=0,(2r-5)(5r-2)=0,r=\frac{5}{2},\frac{2}{5}\,$

Therefore for $r=\frac{5}{2}\,$ the numbers are $\frac{2}{5},1,\frac{5}{2}\,$

For $r=\frac{2}{5}\,$ the numbers are $\frac{5}{2},1,\frac{2}{5}\,$