Question

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

Answer 1

Let three numbers in G.P. are $\frac{a}{r},a,ar$

Here,

$\frac{a}{r}\times a\times ar=729$

$\Rightarrow a^3=729$

$\Rightarrow a=9$

And $(\frac{a}{r}\times a)+(a\times ar)+(\frac{a}{r}\times ar)=819$

$\Rightarrow \frac{81}{r}+81r+81=819$

$\Rightarrow \frac{9}{r}+9r+9=91$

$\Rightarrow 9+9r^2+9=91$

$\Rightarrow 9r^2-82r+9=0$

$\Rightarrow 9r^2-81r-r+9=0$

$\Rightarrow 9r(r-9)-1(r-9)=0$

$r=9,\frac{1}{9}$

Hence, putting the value of a and r, we get the number to be 81, 9 and 1 or 1, 9 and 81.