Gujarat BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions3 Marks
Question
Insert $5$ geometric means between $16$ and $\frac{1}{4}.$
✓
Answer
5 Geometric means between 16 and $\frac{1}{4}$. Let $G_1, G_2, G_3, G_4, G_5$ be 5 geometric means between $a=16$ and $b=\frac{1}{4}$ Then, $16, \mathrm{G}_1, \mathrm{G}_2, \mathrm{G}_3, \mathrm{G}_4, \mathrm{G}_5 \frac{1}{4}$ is a G.P. with $\mathrm{a}=16, \mathrm{~b}=\frac{1}{4}$
$\text{r}=\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{1}{\text{n}+1}}$
$=\Bigg(\frac{\frac{1}{4}}{16}\Bigg)^{\frac{1}{5+1}}=\Big(\frac{1}{26}\Big)^{\frac{1}{6}}=\Big(\frac{1}{2}\Big)$
$\therefore\text{G}_1=\text{ar}=16\times\Big(\frac12\Big)=8$
$\text{G}_2=\text{ar}^2=16\times\frac{1}{4}=4$
$\text{G}_3=\text{ar}^3=16\times\frac{1}{8}=2$
$\text{G}_4=\text{ar}^4=16\times\frac{1}{16}=1$
$\text{G}_5=\text{ar}^5=16\times\frac{1}{{2}^5}=\frac{1}{2}$
Hence, $8,4,2,1,\frac{1}{2}$ are 5 geometric means between 16 and $\frac{1}{4}.$
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