Insertion of GP's between 2 Numbers

Q. If n geometric means between a and b be $G_1,G_2,.......G_n$ and a geometric mean be G, then the true relation is

1. $G_1.G_2.......G_n=G$
2. $G_1.G_2.......G_n=G^{1/n}$
3. $G_1.G_2.......G_n=G^n$
4. $G_1.G_2.......G_n=G^{2/n}$

Q. Let $a_1,a_2,a_3,\dots$ be a G.P. with $a_1=a$ and common ratio $r,$ where $a$ and $r$ positive integers, then the number of ordered pairs $(a,r)$ such that $\sum _{k=1}^{12}{\log }_8{a}_k=2010$ is

Q. If $m$ is the arithmetic mean of two distinct real numbers $l$ and $n(l,n>1)$ and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n,$ then $G_1^4+2G_2^4+G_3^4$ equals

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $4l{m}^{2}n$

Q. If m is the AM of two distinct real numbers l and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals

1. $4l^{2}mn$
2. $4l{m}^{2}n$
3. $lm{n}^2$
4. $l^2{m}^2{n}^2$

Q.

The two numbers between $\frac{1}{16},\frac{1}{6}$ such that first three may be in G.P. and the last three in H.P. are respectively.

1. $\frac{1}{12},\frac{1}{9}$
2. $-\frac{1}{4},-1$
3. $\frac{-1}{12},\frac{-1}{9}$
4. $\frac{1}{4},1$

Q. The two geometric means between the number 1 and 64 are
[Kerala (Engg.) 2002]

1. 1 and 64

2. 4 and 16

3. 2 and 16

4. 8 and 16

Q.

The two numbers between $\frac{1}{16},\frac{1}{6}$ such that first three may be in G.P. and the last three in H.P. are respectively.

1. $\frac{1}{12},\frac{1}{9}$
2. $-\frac{1}{4},-1$
3. $\frac{-1}{12},\frac{-1}{9}$
4. $\frac{1}{4},1$

Q. If three geometric means be inserted between 2 and 32, then the third geometric mean will be

1. 8

2. 4

3. 16
4. 12