Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions5 Marks
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, ar2 $\therefore$ G.P. is 45, 15, 5 or 5, 15, 45
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