Find three numbers in G.P. whose sum is 65 and whose product is 3… - Vidyadip

Question

Find three numbers in G.P. whose sum is $65$ and whose product is $3375$.

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Answer

Let the three number in G.P. be $\frac{\text{a}}{\text{r}},\text{a},\text{ar}$
Sum of these numbers = $\frac{\text{a}}{\text{r}}+\text{a}+\text{ar}=65$
$3375$ = Product of these numbers
$3375=\Big(\frac{\text{a}}{\text{r}}\Big)(\text{a})(\text{ar})=\text{a}^3$
$\text{a}^3=(5)^3\times(3)^3=(15)^3$
$\Rightarrow\text{a}=15$
$\text{a}\Big(\frac{1}{\text{r}}+1+\text{r}\Big)=65$
$15\Big(\frac{1}{\text{r}}+1+\text{r}\Big)=\frac{65}{15}=\frac{13}{3}$
$3+3\text{r}+3\text{r}^2=13\text{r}$
$3\text{r}^2-10\text{r}+3=0$
$3\text{r}^2-\text{r}-9\text{r}+3=0$
$\text{r}(3\text{r}-1)-3(3\text{r}-1)=0$
$\text{r}=3,\frac{1}{3}\ \text{r}=\frac{1}{3}\text{ or}\text{ r}=3$
$\therefore$ G.P. is $a, ar, ar^2$
$\therefore$ G.P. is $45, 15, 5$ or $5, 15, 45$

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