Geometric Mean

Q.

The ratio of the A.M and G.M of two positive numbers a and b, is m : n. Show that $a:b=(m+\sqrt{m^2-n^2}):(m-\sqrt{m^2-n^2})$.

Q. The geometric mean of $6$ and $24$ is

1. $10$
2. $12$
3. $14$
4. $16$

Q.

Find the value of n so that $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the geometric mean between a and b.

Q.

The three arithmetic mean between - 2 and 10 are:

1. 1, 4 and 7

2. -1, 3 and 8

3. 2, 5 and 8

4. -1, 3 and 7

Q. If the geometric mean between a and b is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n},$ then the value of n is

1. 1/2

2. –1/2

3. 2

4. 1

Q.

The ratio of the AM and GM of two positive numbers a and b is m : n, show that $a:b=[m+\sqrt{(m^2-n^2)}]:[m-\sqrt{(m^2-n^2)}].$

Q. Let $a,b,c$ be in G.P. If $x,y$ are arithmetic means between $a,b$ and $b,c$ respectively, then which of the following is/are correct?

1. $\frac{1}{x}+\frac{1}{y}=\frac{1}{b}$
2. $\frac{1}{x}+\frac{1}{y}=\frac{2}{b}$
3. $\frac{a}{x}+\frac{c}{y}=1$
4. $\frac{a}{x}+\frac{c}{y}=2$

Q. The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio .

Q. The sum of 7 numbers in geometric progression is 108. The sum of their reciprocals is 12. The geometric mean of 3 middle terms of the geometric progression is :

Q.

If the AM and GM of two positive numbers a and b are in the ratio m:n show that
$a:b=(m+\sqrt{m^2-n^2}):(m-\sqrt{m^2-n^2}).$

Q. If n geometric means be inserted between a and b then the $n^{th}$ geometric mean will be

1. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n-1)}}$
2. $a{\left(\frac{b}{a}\right)}^{\frac{n-1}{\left(n\right)}}$
3. $a{\left(\frac{b}{a}\right)}^{\frac{1}{\left(n\right)}}$
4. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n+1)}}$

Q. Find the value of n so that may be the geometric mean between a and b .

Q.

If a b and c are in GP and $a^{1/x}=b^{1/y}=c^{1/z}$, prove that x, y and z are in AP.

Q. 7. If the mean of 5, 9, x, 7, 4 and y is 8, then relation between x and y is

Q. The ratio of the A.M and G.M. of two positive numbers a and b , is m : n . Show that .

Q. The arithmetic mean of two positive numbers is $2$ more than its geometric mean. If their difference is $8,$ then the numbers are

1. $10,2$
2. $9,1$
3. $8,3$
4. $12,4$

Q. 18. If (2n)! /3!(2n-3)! And n! /2!(n-2)! Are in the ratio 44:3, then n

Q. The geometric mean of $2,6,8,24$ is

1. $\sqrt{48}=4\sqrt{3}$
2. $16$
3. $24$
4. $36$

Q. The expression $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}is[a\ne b\ne 0]$ is (where a and b are unequal non-zero numbers)

1. A.M. between a and b if n = -1
2. G.M. between a and b$=-\frac{1}{2}$
3. H.M. between a and b if n = 0
4. all are correct

Q. Let $p,p_1$ be A.M. and G.M. between $a$ and $b$ respectively and $q,q_1$ be the A.M. and G.M. between $b$ and $c$ respectively where $a,b,c>0$. If $a,b,c$ are in A.P., then which of the following is CORRECT?

1. $p^2-q^2=q_1^2-p_1^2$
2. $p^2-q^2=p_1^2-q_1^2$
3. $p^2+q^2=p_1^2-q_1^2$
4. $p^2-q^2=p_1^2+q_1^2$

Q. 8. find the mean of observation

Q. Two sequences cannot be in both A.P. and G.P. together.

1. True
2. False

Q. If $G$ is the geometric mean of $x$ and $y,$ then $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}=$

1. $G^2$
2. $\frac{1}{G^2}$
3. $\frac{2}{G^2}$
4. $3G^2$

Q. 6 Mean of twenty observations is 15 if two observation 17 and 20 replaced by 8and9 respectively, then the new mean will be

Q. If the A.M. and G.M. between two numbers are in ratio of $m:n,$ then the ratio of numbers can be

1. $\frac{m+\sqrt{m^2-n^2}}{m-\sqrt{m^2-n^2}}$
2. $\frac{m+\sqrt{m^2-n^2}}{n-\sqrt{m^2-n^2}}$
3. $\frac{m+\sqrt{m^2+n^2}}{m-\sqrt{m^2+n^2}}$
4. $\frac{n+\sqrt{m^2-n^2}}{m-\sqrt{m^2-n^2}}$

Q. If a, b, c and d are in G.P. show that .

Q. lf $1,1,\alpha$ are the roots of $x^3-6x^2+9x-4=0$, then $\alpha =$

1. $4$
2. $-3$
3. $3$
4. $-4$

Q. If x is the arithmetic mean of y and z and the two geometric means between y and z are $G_1\mathrm{and}{G}_2$, then $G_1^3+G_2^3=$.

1. $\mathrm{xyz}$
2. $\frac{1}{\mathrm{xyz}}$
3. $2\mathrm{xyz}$
4. $\frac{2}{\mathrm{xyz}}$

Q. The product of three geometric mean between $\frac{1}{4}$ and $4$ is

1. $1$
2. $14$
3. $28$
4. $32$

Q. If geometric mean between x and y is G, then the value of $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}$is equal to

1. $G^2$
2. $\frac{1}{G^2}$
3. $2G^2$
4. $\frac{3}{G^2}$