Selecting Consecutive Terms in GP

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. In a G.P. if $t_3=2$ and $t_6=-\frac{1}{4},$ then $t_{10}=$

1. $-\frac{1}{128}$
2. $\frac{1}{128}$
3. $-\frac{1}{64}$
4. $\frac{1}{64}$

Q. If $f\left(x\right)=\underset{n\to \infty }{lim}(2x+4x^3+\dots +2^n{x}^{2n-1})$, where $x\in (0,\frac{1}{\sqrt{2}}),$ then $\int f\left(x\right)dx$ is equal to

1. $\log \left(\frac{1}{\sqrt{1+2x^2}}\right)+C$
2. $\log \sqrt{1-2x^2}+C$
3. $\log \left(\frac{1}{\sqrt{1-2x^2}}\right)+C$
4. $\log (1-2x^2)+C$

Q. Let the digits of a three digit number are in G.P. If $400$ is subracted from the number and the digits of the new number are in A.P., then the last digit of the original number is

1. $8$
2. $9$
3. $1$
4. $2$

Q. Let $a,b,c$ be three distinct real numbers in geometric progression. If $x$ is real and $a+b+c=xb,$ then $x$ can be

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

Q. The first term of a G.P. is $1.$ If the sum of the third and the fifth term is $90,$ then the common ratio is

1. $-\sqrt{10}$
2. $\sqrt{10}$
3. $-3$
4. $3$

Q. Let $a_n$ be the the $n^{th}$ term of a G.P. of positive numbers such that $\sum _{n=1}^{100}{a}_{2n}=\alpha$ and $\sum _{n=1}^{100}{a}_{2n-1}=\beta ,\alpha \ne \beta .$ Then the common ratio of G.P. is

1. $\frac{\alpha }{\beta }$
2. $\frac{\beta }{\alpha }$
3. $\sqrt{\frac{\alpha }{\beta }}$
4. $\sqrt{\frac{\beta }{\alpha }}$

Q. Three numbers are in G.P. If we double the middle term, then they will be in A.P. The common ratio of the G.P. is

1. $1\pm \sqrt{3}$
2. $2\pm \sqrt{3}$
3. $1\pm \sqrt{5}$
4. $2\pm \sqrt{5}$

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. $\sqrt{xy}$

Q. Three positive 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. $2-\sqrt{3}$
2. $2+\sqrt{3}$
3. $\sqrt{3}-2$
4. $\sqrt{3}$

Q. Three positive numbers form 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. $3+\sqrt{2}$
3. $2+\sqrt{3}$
4. $\sqrt{2}+\sqrt{3}$

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

1. $xϵ[\frac{3}{4},\infty ]$
2. $xϵR^{+}$
3. $xϵ(-1,\infty )$
4. none of these