Maharashtra BoardEnglish MediumSTD 11 ScienceMathsGeometric Progressions1 Mark
Question
Write the product of n geometric means between two number a and b.
✓
Answer
Let $\text{x}_1, \text{x}_2, \text{x}_3 \dots\text{x}_\text{n}$ be n G.M. between a and b then
$\text{a},\text{x}_1, \text{x}_2, \text{x}_3\dots \text{x}_\text{n},\text{b}$
$\text{b}=\text{a}_{\text{x}+2}=\text{ar}^{\text{n}+1}$
$\text{r}=\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{1}{\text{n}+1}}\cdots(1)$
$\text{x}_1=\text{a}_2=\text{ar}$
$\text{x}_2=\text{a}_3=\text{ar}^2$
$\text{x}_\text{n}=\text{a}_{\text{n}+1}=\text{ar}^{\text{n}}$
Product of n GM
$=\text{x}_1,\text{x}_2\dots\text{x}_\text{n}$
$=(\text{ar})\big(\text{ar}^2\big)\dots\big(\text{ar}^\text{n}\big)$
$=\text{a}^{\text{n}}\text{r}^{1+2+\dots+\text{n}}$
$=\text{a}^\text{n}\text{r}\frac{\text{n}(\text{n}+1)}{2}$
$=\text{a}^\text{n}\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{\text{n}}{2}}$ [Using equation (1)]
$=\text{a}^{\text{n}}\frac{\text{b}^{\frac{\text{n}}{2}}}{\text{a}^{\frac{\text{n}}{2}}}$
$=\text{a}^{\frac{\text{n}}{2}}\text{b}^{\frac{\text{n}}{2}}$
$=(\text{ab})^{\frac{\text{n}}{2}}$
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