Selecting Consecutive Terms in GP

Q.

If the product of three numbers in GP is 216 and the sum of their products in pairs is 156, find the numbers.

Q.

For a, b, c to be in GP, what is the value of $\frac{a-b}{b-c}$

Q.

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.

Q. If x, $G_1,G_2,$ y be the consecutive terms of a G.P., then the value of $G_1{G}_2$ will be

1. xy
2. $\sqrt{xy}$
3. $\frac{y}{x}$
4. $\frac{x}{y}$

Q.

If S be the sum, P be the product and R be the sum of the reciprocals of n terms in a GP, prove that $P^2={\left(\frac{S}{R}\right)}^n$

Q. The sum of first three terms of a G.P. is and their product is 1. Find the common ratio and the terms.

Q. Find the ${10}^{th}$ term and sum of the 10 terms of the series $1,2,2^2,2^3,2^4....$

1. 256, 512

2. 512, 1023

3. 512, 1024

4. 256, 511

Q. Let $n\ge 1$ be a positive integer. Suppose, between two distinct positive real numbers $k$ and $l,n$ arithmetic means $a_1,a_2,...,a_n$ and $n$ harmonic means $h_1,h_2,....,h_n$ are inserted. For $i=1,2,....,n$ and $j=1,2,....,n$, let $p(i,j)=a_i{h}_j$ then

1. If $n$ is even, there are an even number of ordered pairs $(i,j)$, such that $p(i,j)=kl$
2. $p(i,j)=kl$ for some $i$ and $j$
3. If $n$ is odd, there are an even number of ordered pairs $(i,j)$, such that $p(i,j)=kl$
4. Whenever $n$ is odd there is an $i$ such that $p(i,i)=kl$

Q. A three digit number whose consecutive digits form a G .P. If we subtract 792 from this number, we get a number consisting of the same digits written in the reverse order. Now if we increase the second digit of the required number by 2, the resulting number will form an A.P. Then the number is-

1. $139$
2. $927$
3. $931$
4. $763$

Q.

Find n, if the ratio of the fifth term from the beginning to fifth term from the end in the expansion of ${(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})}^n$ is $\sqrt{6}:1.$

Or

Prove that the coeffficient of the middle term in the expansion of $(1+x{)}^{2n}$ is equal to the sum of the coefficient of middle terms in the expansion of $(1+x{)}^{2n-1}.$

Q. If the product of three consecutive terms of G.P. is 216 and the sum of product of pair-wise is 156, then the numbers will be
[MNR 1978]

1. 1, 3, 9

2. 2, 6, 18

3. 3, 9, 27
4. 2, 4, 8

Q. If $x,2x+2$ and $3x+3$ are in G.P. then the next term in this sequence is

1. $27$
2. $-27$
3. $-\frac{27}{2}$
4. $\frac{27}{2}$

Q. If a, b, c, d are in G.P., then show that
i) $(a+b{)}^2,(b+c{)}^2,(c+d{)}^2$ are in G.P
ii) $\frac{1}{a^2+b^2},\frac{1}{b^2+c^2},\frac{1}{c^2+d^2}$ are in G.P

Q.

If a, b, c and d are in G.P show that $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd{)}^2.$

Q. If three distinct real numbers a, b, c are in G.P and $a+b+c=ax$, then

1. x ∊ (-1, ∞)
2. none of these
3. 
4. 

Q.

Find the sum of the GP \((1+x)^{21}+(1+x)^{22}+......(1+x)^{30}

1. 

2. 

3. 

4. 

Q. If the product of three consecutive terms of G.P. is 216 and the sum of product of pair-wise is 156, then the numbers will be
[MNR 1978]

1. 2, 4, 8

2. 1, 3, 9

3. 2, 6, 18

4. 3, 9, 27

Q.

If a, b, c, d are in GP, prove that

$(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd{)}^2$

Q. $x$ is one of the three consecutive numbers in G.P. whose sum is 38 and their product is 1728. The value of $x$ can be ___.

1. 18
2. 12
3. 4
4. 16

Q. The product of three consecutive terms of a $\text{G.P.}$ is $512.$ If $4$ is added to each of the first and the second of these terms, then the three terms now form an $\text{A.P.}$ Then the sum of the original three terms of the given $\text{G.P.}$ is:

1. $36$
2. $32$
3. $28$
4. $24$

Q.

If the arithmetic mean of two numbers be A and geometric mean be G, then the numbers will be

1. 

2. 

3. 

4. 

Q. Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:

1. $2-\sqrt{3}$
2. $2+\sqrt{3}$
3. $\sqrt{2}+\sqrt{3}$
4. $3+\sqrt{2}$

Q.

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

1. 

2. 

3. 

4. 

Q. If x, $G_1,G_2,$ y be the consecutive terms of a G.P., then the value of $G_1{G}_2$ will be

1. $\frac{y}{x}$
2. $\frac{x}{y}$
3. xy
4. $\sqrt{xy}$