Question

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.

Answer 1

Let the numbers are $\frac{a}{r},a$ and ar. Then

We have

$\frac{a}{r}+a+ar=\frac{39}{10}$

And

$\frac{a}{r}.a.ar=1$

$a^3=1$

$a=1$

Now we have

$\frac{1}{r}+1+r=\frac{39}{10}$

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

$r^2-\frac{29}{10}r+1=0$

$10r^2-29r+10=0$

$10r^2-25r+4r=10=0$

$5r(2r-5)-2(2r-5)=0$

(2r - 5)(5r - 2) = 0

$r=\frac{5}{2},\frac{2}{5}$

Hence, putting the value of a and r, the required number are $\frac{2}{5},1,\frac{5}{2}or\frac{5}{2},1,\frac{2}{5}$