Geometric Progression

Q.

Sum of $n$ terms of series $12+16+24+40+\cdots$will be

1. $2\left(2^n-1\right)+8n$

2. $2\left(2^n-1\right)+6n$

3. $3\left(2^n-1\right)+8n$

4. $4\left(2^n-1\right)+8n$

Q. If $G_1,G_2,G_3....G_n$ are n numbers inserted between the numbers a and b of a G.P. then $G_1\times G_2\times G_3\times ....G_n=$

1. $\sqrt{ab}$
2. ${\left(\sqrt{ab}\right)}^n$
3. ${\left(\sqrt{ab}\right)}^{n+1}$
4. $(\frac{a+b}{2}{)}^n$

Q. If $(k-3),(2k+1)$ and $(4k+3)$ are three consecutive terms of an A.P., find the value of k.

Q. In the quadratic equation $a{x}^2+bx+c=0,\Delta =b^2-4acand\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3,$ are in G.P. where $\alpha ,\beta$ are the root of $a{x}^2+bx+c=0,$ then

1. $b\Delta =0$
2. $\Delta \ne 0$
3. $c\Delta =0$
4. $\Delta =0$

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)th term of G.P. is m and (p-q)th term is n, then pth term will be

1. $\sqrt{mn}$

2. $\left(\frac{m}{n}\right)$
   

3. $(m-n)$
   

4. $mn$
   

Q.

If a>0, b>0, c>0 are in G.P., then $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ are in

1. H.P.
   

2. A.P.
   

3. G.P.
   

4. None of these

Q. If ${(a+\frac{1}{a})}^2=3$ and $a\ne 0$; then show :
$a^3+\frac{1}{a^3}=0$.

Q. If the roots of the cubic equation $a{x}^3+b{x}^2+cx+d=0$ are in G.P. then

1. $c^{3}a=b^{3}d$
   
2. $a^{3}b=c^{3}d$
   
3. $a{b}^3=c{d}^3$
   
4. $c{a}^3=b{d}^3$
   

Q. Express as a continued fraction the series
$\frac{1}{a_0}-\frac{x}{a_0{a}_1}+\frac{x^2}{a_0{a}_1{a}_2}-\cdots +{(-1)}^n\frac{x^n}{a_0{a}_1{a}_2\dots a_n}$.

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. An A.P., a G.P., and a H.P. have a and b for their first two terms their $(n+2{)}^{th}$ terms will be in G.P. if $\frac{b^{2n+2}-a^{2n+2}}{ab(b^{2n}-a^{2n})}$

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

Q. Let $\alpha ,\beta$ be the roots of $x^2-x+p=0and\gamma ,\delta$ be the roots of $x^2-4x+q=0.$~If $\alpha ,\beta ,\gamma ,\delta ,$ are in G.P., then the integral values of p and q respectively, are

1. -2, -32
2. -2, 3
3. -6, 3
4. -6, -32

Q.

If x, y, z are in G.P. and $a^x$ = $b^y$ = $c^z$, then

1. 

2. 

3. 

4. 

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

Q. write this in the exponential form ^{power 11} root 3000

Q. Consider the following statements. For any integer n.
I. $n^2+3$ is never divisible by $17$.
II. $n^2+4$ is never divisible by $17$. then.

1. Both I and II are true
2. I is false and II is true
3. Both I and II are false
4. I is true and II is false

Q. Which of the following is a monomial?

1. $4x^2$
2. $4x^2+2x^2$
3. $x^3+4x^2+2$
4. $4x^2+4x+2$

Q. Suppose the sequence $a_1,a_2,a_3.....$ is an arithmetic progression of distinct numbers such that the $a_1,a_2,a_4,a_6.....$is a geometric progression. The common ration of the geometric progression is:

1. not determinable
2. $-2$
3. $4$
4. $a_1$

Q. If ${}^8{P}_r{=}^8{P}_{r+1}$, then find value of $r$.

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$

1. ${\alpha }^3+{\beta }^3=\frac{3abc-b^3}{a^3}$
2. ${\alpha }^3+{\beta }^3=\frac{b^3-3abc}{a^3}$
3. ${\alpha }^2+{\beta }^2=\frac{2ac-b^2}{a^2}$
4. $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{b^2-2ac}{c^2}$

Q. The coefficient of the term -32$x^{29}$ = ​​​​​​.

1. 32
2. 29
3. 0
4. -32