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 $\frac{a}{r},a,ar$ be the first three terms of the G.P.

$\frac{a}{r}+a+ar=\frac{39}{10}$ ….(1)

$(\frac{a}{r})(a)(ar)=1$ ...(2)

From (2), we obtain

$a^3=1$

⇒ a = 1 (Considering real roots only)

Substituting a = 1 in equation (1), we obtain

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

⇒ 1 + r + $r^2=\frac{39}{10}{r}$

⇒ 10 + 10r + $10r^2-39r=0$

⇒ 10$r^2-29$r + 10 = 0

⇒ 10$r^2$ - 25r - 4r + 10 = 0

⇒ 5r (2r - 5) - 2 (2r - 5) = 0

⇒ (5 - r)(2r - 5) = 0

⇒ r = $\frac {2} {5} or \frac {5} {2}$

Thus, the three terms of G.P. are $\frac{5} {2},1,and \frac{2} {5}.$