Geometric Progression

Q. Let $a,b,c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a,b,c$ are in geometric progression and the arithmetic mean of $a,b,c$ is $b+2$, then the value of $\frac{a^2+a-14}{a+1}$ is

Q. If $a$, $b$ and $c$ be three distinct real numbers in G.P. and $a+b+c=xb$, then $x$ cannot be :

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

Q. If $\left\{x\right\}$ and $\left[x\right]$ represent the fractional and the integral part of $x$ respectively, then $\frac{2019}{2020}\left[x\right]+\frac{x}{2020}+\sum _{r=1}^{2019}\frac{\{x+r\}}{2020}$ is equal to

1. $x$
2. $1010x$
3. $2021x$
4. $\frac{1}{2021}x$

Q.

The sum of all positive divisors of $960$ is?

1. $3048$

2. $3087$

3. $3047$

4. $2180$

Q. If $x,y$ and $z$ are $p^{th},q^{th}$ and $r^{th}$ terms respectively of an A.P. and also of a G.P., then $x^{y-z}{y}^{z-x}{z}^{x-y}$ is equal to

1. $0$
2. $1$
3. None of these
4. $xyz$

Q.

If three unequal positive real numbers a, b, c are in G.P. and a-b, c-a, a-b are in H.P., then the values of a+b+c is independent of

1. a

2. b

3. c

4. None of these

Q.

If $1,2,3$ and $4$ are the roots of the equation $x^4+a{x}^3+b{x}^2+cx+d=0$ then $a+2b+c$ is equal to

1. $-25$

2. $0$

3. $10$

4. $24$

Q. If $(p+q{)}^{\text{th}}$ term of a G.P. is ${}^{\prime }{a}^{\prime }$ and its $(p-q{)}^{\text{th}}$ term is ${}^{\prime }{b}^{\prime }$ where $a,b\in R^{+}$, then its $p^{\text{th}}$ term is

1. $\sqrt{\frac{a^3}{b}}$
2. $\sqrt{\frac{b^3}{a}}$
3. $\sqrt{ab}$
4. None of these

Q. The sum of first 20 terms of the sequence 0.5, 0.55, 0.555, ...., is ___.

1. $\frac{5}{81}(179+{10}^{-20})$
2. $\frac{5}{9}(99+{10}^{-20})$
3. $\frac{5}{81}(179-{10}^{-20})$
4. $\frac{5}{9}(99-{10}^{-20})$

Q. If the sum of the first three terms of a G.P. is $\frac{13}{12}$ and their product is $-1,$ then which of the following can be terms of those G.P.?

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

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

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

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

Q. The sum of first $20$ terms of the sequence $0.7,0.77,0.777,...,$ is

1. $\frac{7}{81}[179-{10}^{-20}]$
2. $\frac{7}{9}[99-{10}^{-20}]$
3. $\frac{7}{81}[179+{10}^{-20}]$
4. $\frac{7}{9}[99+{10}^{-20}]$

Q. If $p,q,r$ are in A.P. and $p^2,q^2,r^2$ are in G.P, where $p<q<r$ and $p+q+r=\frac{3}{2},$ then the value of $p$ is

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

Q. The fourth, seventh and the last term of a G.P. are $10,80$ and $2560$ respectively. Then

1. first term is $\frac{5}{2}$
2. common ratio is $2$
3. number of terms is $12$
4. ${10}^{th}$ term is $640$

Q. If $T_{n+1}=\frac{2T_n+1}{2},n\in N$ and $T_{10}=\frac{19}{2}$, then the ${101}^{th}$ term of the sequence is

1. $50$
2. $52$
3. $54$
4. $55$

Q. If $a=b^2=c^4=d^8=e^{16}\ne 1$, where $b,c,d,e>0,$ then the value of ${\log }_{cde}abc$ is

1. equal to zero
2. greater than $4$
3. less than $1$
4. greater than $1$

Q. The sum of first $8$ terms of the series $3,6,12,24,\dots$ is

1. $452$
2. $541$
3. $689$
4. $765$

Q. The $n^{th}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1$.

Using the above information $a_n$ will be

1. $a_n=n^2+n+1$
2. $a_n=n^2-n+1$
3. $a_n=n^2-n-1$
4. $a_n=n^2+n-1$

Q. If $x=1+a+a^2+....\infty$ and $y=1+b+b^2+....\infty ,$ then the value of $1+ab++a^2{b}^2+....\infty$ is
$(0<a<1,0<b<1)$

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

Q. The greatest positive integer $k$, for which ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\cdots +{49}^2+49+1,$ is

1. $32$
2. $60$
3. $65$
4. $63$

Q. Let $S$ denotes the sum of series $\frac{3}{2^3}+\frac{4}{2^4\cdot 3}+\frac{5}{2^6\cdot 3}+\frac{6}{2^7\cdot 5}+\cdots +\infty .$ Then the of $S^{-1}$ is

Q. Let $a_n$ be the $n^{\text{th}}$ 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. The difference between fourth and first term of a $G.P.$ is $52$ and sum of the first three terms is $26$. If $T_n$ and $S_n$ represent $n^{th}$ term and sum to $n$ terms respectively of the $G.P.$, then which of the following is/are true

1. $T_7=486$
2. $T_7=1458$
3. $S_6=972$
4. $S_6=728$

Q. If $x,2y,3z$ are in A.P. where the distinct numbers $x,y,z$ are in G.P, then the common ratio of G.P. is

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

Q. The sum of an infinite geometric series with positive terms is $3$ and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is :

1. $\frac{4}{9}$
2. $\frac{2}{9}$
3. $\frac{1}{3}$
4. $\frac{2}{3}$

Q. In a geometric sequence, the first term is a and the common ratio is r. If $S_n$ denotes the sum to n, terms and $U_n={\sum }_{n=1}^n{S}_n$
then $r{S}_n+(1-r)U_n=$

1. n
2. $n^{2}a$
3. na
4. None of these

Q. The sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, ....., is ___.

1. $\frac{7}{81}(179-{10}^{-20})$
2. $\frac{7}{9}(99-{10}^{-20})$
3. $\frac{7}{81}(179+{10}^{-20})$
4. $\frac{7}{9}(99+{10}^{-20})$

Q. Let $\alpha ,\beta ,\gamma$ be in AP and $x,y,z$ be in GP. If $\tan \alpha =x,\tan \beta =y$ and $\tan \gamma =z,$ then

1. $x=y=z$
2. $xz=1$
3. $x=y=z\ne 1$
4. $x,y,z$ are in AP

Q.

If $p^{th}$, $q^{th}$, $r^{th}$ and $s^{th}$ terms of an A.P. be in G.P., then (p - q), (q - r), (r - s) will be in

1. G.P

2. A.P

3. H.P

4. None of these