Question

The sum of four numbers in G.P. is $820$ and their product is $5,31,441$. Find the numbers.

Answer 1

Sum of all numbers = 820

$\frac{a}{r^3}+\frac{a}{r}+ar+a{r}^3=820$

$a[\frac{1}{r^3}+\frac{1}{r}+r+r^3]=820$...(1)

Product of all numbers = 531441

$\frac{a}{r^3}\times \frac{a}{r}\times ar\times a{r}^3=531441$

$a^4=531441\Rightarrow a=27$

From(1)

$27[\frac{1}{r^3}+\frac{1}{r}+r+r^3]=820$

Let $r=3$

$\therefore 27[\frac{1}{3^3}+\frac{1}{3}+3+3^3]=820$

$\therefore r=3$ is correct

$\therefore$ Numbers are:

$\frac{a}{r^3}=1$

$\frac{a}{r}=9$

$ar=81$

$a{r}^3=729$