A.G.P

Q.

What are AP, GP, and HP?

Q.

What is the difference between arithmetic and geometric progression?

Q. Three numbers are in an increasing geometric progression with common ratio $r.$ If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d.$ If the fourth term of GP is $3r^2,$ then $r^2-d$ is equal to

1. $7-7\sqrt{3}$
2. $7+\sqrt{3}$
3. $7-\sqrt{3}$
4. $7+3\sqrt{3}$

Q. The value of $2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}}\cdot 8^{\frac{1}{16}}\cdot {16}^{\frac{1}{32}}\dots \dots$ is

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

Q. The sum of the series $1+3x+5x^2+7x^3+\dots$ upto $n$ terms is

1. $\frac{1}{1+x}+2n\frac{(1+x^{n-1})}{(1+x^2)}-\frac{(2n-1)x^n}{1+x}$
2. $\frac{1}{1-x}+2x\frac{(1-x^{n-1})}{(1-x{)}^2}-\frac{(2n-1)x^n}{(1-x)}$
3. $\frac{1}{1-x}-\frac{1-x^{n-1}}{(1-x{)}^2}-\frac{(2n-1)x^n}{(1-x)}$
4. $\frac{1}{1-x}-2x\frac{(1-x{)}^{n-1}}{(1-x{)}^2}-\frac{x^n}{(1-x)}$

Q.

If ‘a, b, c’ are in arithmetic progression $b-a,c-b\&a$ are in geometric progression then $a:b:c$ is

1. $1:2:3$

2. $1:3:5$

3. $2:3:4$

4. $1:2:4$

Q.

If$‘a,b\&c’$ are in Arithmetic Progression and $a^2,b^2,c^2$are in Harmonic Progression, then

1. $a\ne b\ne c$

2. $a^2=b^2=\frac{c^2}{2}$

3. $‘a,b\&c’$ are in G, P

4. $\frac{-a}{2},b,c$ are in G.P

Q.

What are the types of progression?

Q. $\text{Let}\left\{a_n\right\}\text{be a sequence such that}{a}_0=1,a_1=0,a_n=3a_{n-1}-2a_{n-2}.\text{Then, which of the following is a correct statement?}$

1. $a_{45}=2^{45}$
2. $a_{51}=2^{51}-2$
3. $a_{49}=\sqrt{2}-\sqrt{2^{49}}$
4. $a_{48}=2(1-2^{47})$

Q. The sum to $50$ terms of the series $1+2(1+\frac{1}{50})+3{(1+\frac{1}{50})}^2+\dots$ is given by

1. $2450$
2. $2500$
3. $2550$
4. None of these

Q. The number of distinct real roots of the equation $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0,$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

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

Q.

If $a,b,c$are in $A.P$ as well as in $G.P.$ then

1. $a=b\ne c$

2. $a\ne b=c$

3. $a\ne b\ne c$

4. $a=b=c$

Q. If $\frac{3}{5}+\frac{5}{15}+\frac{7}{45}+\frac{9}{135}+\dots =\frac{a}{b},$ then
(where $a$ and $b$ are coprime)

1. $a=6$
2. $b=5$
3. $a=5$
4. $b=6$

Q. $\text{Let a, b, c are non-zero real numbers such that}\frac{1}{a},\frac{1}{b},\frac{1}{c}\text{are in arithmetic progression}\text{and a, b, -2c are in geometric progression, then which of the following statement(s) must be true?}$

1. $a^2,b^2,4c^2\text{are in geometric progression}$
2. $-2a,b,-2c\text{are in arithmetic progression}$
3. $a^3+b^3+c^3-3abc=0$
4. $a^2,b^2,c^2\text{are in harmonic progression}$

Q.

If $\frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in HP, then

1. $a^{2}b,c^{2}a,b^{2}c$ are in AP

2. $a^{2}b,b^{2}c,c^{2}a$ are in HP

3. $a^{2}b,b^{2}c,c^{2}a$ are in GP

4. None of these

Q. If positive square root of $a^{\frac{1}{a}}.(2a{)}^{\frac{1}{2a}}.(4a{)}^{\frac{1}{4a}}.(8a{)}^{\frac{1}{8a}}\dots \infty$ is $\frac{8}{27},$ then the value of $a$ is

1. $\frac{2}{3}$
2. $\frac{1}{5}$
3. $\frac{1}{3}$
4. $\frac{3}{4}$

Q. If sum of first $n$ terms of an A.P. is $S_n=3n^2-2n,$ then the value of $\sum _{n=1}^{\infty }\left(\frac{21}{(S_n{S}_{n+2}+S_{n-1}{S}_{n+1})-(S_n{S}_{n+1}+S_{n-1}{S}_{n+2})}\right)$ is equal to

Q.

If $a,b,c$ are in $A.P$. and $a^2,b^2,c^2$ are in $H.P$., then

1. $a=b=c$

2. $2b=3a+c$

3. $b^2=\sqrt{\frac{ac}{8}}$

4. None of these

Q. The value of the summation $\sum _{i=1}^{\infty }\frac{2i-1}{2^{i+1}}$ is

1. $\frac{4}{3}$
2. $\frac{3}{2}$
3. $3$
4. $\frac{5}{3}$

Q. If $2A+B^T=\left[\begin{array}{cc}2& -3\\ 4& 7\end{array}\right]$ and $A^T-B=\left[\begin{array}{cc}4& -5\\ 0& 1\end{array}\right]$, then $A$ is equal to

1. $\frac{1}{3}\left[\begin{array}{cc}6& -3\\ -1& 8\end{array}\right]$
2. $\left[\begin{array}{cc}2& -3\\ -1& 8\end{array}\right]$
3. $\frac{1}{2}\left[\begin{array}{cc}2& 3\\ -1& 8\end{array}\right]$
4. $\left[\begin{array}{cc}3& -3\\ -1& 8\end{array}\right]$

Q. The sum to (n + 1) terms of the series $\frac{c_0}{2}-\frac{c_1}{3}+\frac{c_2}{4}-\frac{c_3}{5}+....is$

1. 
2. 
3. 
4. 

Q. The value of $\sum _{n=0}^{\infty }\frac{2n+3}{3^n}$ is equal to

Q. If $S=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{3^4}+\cdots \infty ,$ then the value of $4S$ is

Q. For any positive integer $n$, let $S_n:(0,\infty )\to R$ be defined by $S_n\left(x\right)=\sum _{k=1}^n{\cot }^{-1}\left(\frac{1+k(k+1)x^2}{x}\right),$ where for any $x\in R,{\cot }^{-1}\left(x\right)\in (0,\pi )$ and ${\tan }^{-1}\left(x\right)\in (-\frac{\pi }{2},\frac{\pi }{2})$. Then which of the following statements is(are) TRUE?

1. $S_{10}\left(x\right)=\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1+11x^2}{10x}\right)$, for all $x>0$
2. $\underset{n\to \infty }{lim}\cot \left(S_n\right(x\left)\right)=x,$ for all $x>0$
3. The equation $S_3\left(x\right)=\frac{\pi }{4}$ has a root in $(0,\infty )$
4. $\tan \left(S_n\right(x\left)\right)\le \frac{1}{2},$ for all $n\ge 1$ and $x>0$

Q.

For the function $f\left(x\right)=e^x,a=0,b=1$, the value of $c$ in mean value theorem will be

1. $\log x$

2. $\log (e–1)$

3. $0$

4. $1$

Q. If $n>3$ and $a,bϵR,$ then the value of $ab-n(a-1)(b-1)+\frac{n(n-1)}{1.2}(a-2)(b-2)-......+(-1{)}^n(a-n\left)\right(b-n)$ is equal to

1. 
2. 
3. 
4. 

Q.

If the P^th, q^th , and r^th tems of an A.P. are in G.P., then common ratio of the G.P. is

1. 

2. 

3. 

4. 

Q. The value of sum $\sum _{n=1}^{\infty }\frac{n}{7^n}$ is

1. $\frac{6}{7}$
2. $\frac{7}{6}$
3. $\frac{49}{36}$
4. $\frac{7}{36}$

Q. $\text{Let}\left\{a_n\right\}\text{be a sequence such that}{a}_0=1,a_1=0,a_n=3a_{n-1}-2a_{n-2}.\text{Then, which of the following is a correct statement?}$

1. $a_{45}=2^{45}$
2. $a_{51}=2^{51}-2$
3. $a_{48}=2(1-2^{47})$
4. $a_{49}=\sqrt{2}-\sqrt{2^{49}}$

Q. If $S_n=\sum _{r=1}^n\sum _{t=0}^{r-1}\left(\frac{1}{6^n}{}^n{C}_r{}^r{C}_t{4}^t\right),$ then the value of $l$ where $l=\sum _{n=1}^{\infty }(1-S_n)$ is

1. $5$
2. $1$
3. $4$
4. $6$