Insert 6 geometric means between 27 and 1 81 . - Vidyadip

Question

Insert $6$ geometric means between $27$ and $\frac{1}{81}.$

✓

Answer

6 Geometric means between 27 and $\frac{1}{81}.$ Let $G_1, G_2, G_3, G_4, G_5, G_6$_ be 6 geometric means between a = 27 and $\text{b}=\frac{1}{81}$ Then, $27, G_1, G_2, G_3, G_4, G_5, G_6, \frac{1}{81}$ is a G.P. with common ratio r given by $\text{r}=\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{1}{\text{n}+1}}$
$=\Bigg(\frac{\frac{1}{81}}{27}\Bigg)^{\frac{1}{6+1}}=\Big(\frac{1}{81\times27}\Big)^{\frac{1}{7}}=\Big(\frac{1}{3^7}\Big)^{\frac{1}{7}}$
$\therefore\text{G}_1=\text{ar}=27\times\Big(\frac13\Big)=9$
$\text{G}_2=\text{ar}^2=27\times\frac{1}{9}=3$
$\text{G}_3=\text{ar}^3=27\times\frac{1}{27}=1$
$\text{G}_4=\text{ar}^4=27\times\frac{1}{27\times3}=\frac13$
$\text{G}_5=\text{ar}^5=27\times\frac{1}{{3}^5}=\frac{1}{9}$
$\text{G}_6=\text{ar}^6=27\times\frac{1}{36}=\frac{1}{27}$ Hence, $9, 3, 1, \frac{1}{3},\frac19,\frac{1}{27}$ are 6 geometric means between 27 and $\frac{1}{81}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from Geometric Progressions→Geometric Progressions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the number of ways in which one can post 5 letters in 7 letter boxes.

Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x - 2y = 1.

In the expansion of $(1+a)^{m+n}$, prove that the coefficients of $a^m$ and $a^n$ are equal.

Two plants A and B of a factory show following results about the number of workers and the wages paid to them:

	Plant A	Plant B
No. of workers	5000	6000
Average monthly wages	₹ 2500	₹ 2500
Variance of distribution of wages	81	100

In which plant A or B is there greater variability in individual wages?

A card is drawn from a deck of 52 cards. Find the probability of getting an ace or a spade card.

Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas: $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$

Find k so that $\lim\limits_{\text{x}\rightarrow2}\text{f(x)}$ may exist, where $\text{f(x)}=\begin{cases}2\text{x}+3, & \text{x}\le 2\\\text{x}+\text{k}, & \text{x} > 2\end{cases}.$

Write the following relation as the sets of ordered pairs:
A relation R on the set {1, 2, 3, 4, 5, 6, 7}defined by $(\text{x, y})\in \text{R}\Leftrightarrow\text{x}$ is relatively prime to y.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always $10 m$ and the distance between the flag posts is $8 m$. Find the equation of the path traced by the man.

Find the standard deviation for the following data:

$x_i$	3	8	13	18	23
$f_i$	7	10	15	10	6