Geometric Progression

Q.

If the ratio of the sum of first three terms and the sum of first six terms of a G.P. be 125 : 152, then the common ratio $r$ is ___.

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

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

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

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

Q.

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

1. 

2. 

3. 

4. 

Q.

If $a,b,c$ are in A.P. and $a,c-b,b-a$ are in G.P., where$a\ne b\ne c$, then $a:b:c$ is ___.

1. 3:2:1

2. 1:3:5

3. 1:2:4

4. 1:2:3

Q.

If $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}$ and $a,b,c$ are in G.P., then which of the following are true?

1. $x,y,z$ will be in A.P.

2. $x,y,z$ will be in G.P.

3. $x,y,z$ will be in H.P.

4. None of the above

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. None of these

4. H.P

Q.

If the numbers be in G.P., then their logarithms will be in

1. A.P

2. H.P

3. None of these

4. G.P

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

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. Sum of the first n terms of the series
$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots$ is equal to

1. $2^n-n-1$
2. $1-2^{-n}$
3. $n+2^{-n}-1$
4. $2^n+1.$