Sum of Infinite Terms of a GP

Q. If $\text{Re}\left(\frac{z-1}{2z+i}\right)=1,$ where $z=x+iy$, then the point $(x,y)$ lies on a

1. circle whose centre is at $(-\frac{1}{2},-\frac{3}{2}).$
2. straight line whose slope is $\frac{3}{2}$.
3. circle whose diameter is $\frac{\sqrt{5}}{2}$.
4. straight line whose slope is $-\frac{2}{3}$.

Q.

The value of $9^{\frac{1}{3}}{.9}^{\frac{1}{9}}{.9}^{\frac{1}{27}}.....$ to $\infty$ is

1. 9

2. 3

3. 1

Q.

For integers n and r, let...$\left(\begin{array}{c}n\\ r\end{array}\right)=\left\{\begin{array}{ll}n{C}_r,& ifn\ge r\ge 0\\ 0,& Otherwise\end{array}\right.$The maximum value of k for which the sum $\sum _{i=0}^k\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}15\\ k-i\end{array}\right)+\sum _{i=0}^{k+1}\left(\begin{array}{c}12\\ i\end{array}\right)\left(\begin{array}{c}13\\ k+1-i\end{array}\right)$ exists is equal to

Q.

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression equals

1. $\frac{\left(1-\sqrt{5}\right)}{2}$

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

3. $\sqrt{5}$

4. $\frac{\left(\sqrt{5}-1\right)}{2}$

Q. The sum of the series $S=1+2\left(\frac{10}{11}\right)+3{\left(\frac{10}{11}\right)}^2+\cdots \text{upto}\infty$ is equal to

1. $121$
2. $120$
3. $111$
4. $110$

Q.

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 192. The common ratio of the original G.P. is

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

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

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

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

Q.

The fractional value of $2.357$ is

1. $\frac{2355}{1001}$

2. $\frac{2379}{997}$

3. $\frac{2355}{999}$

4. none of these

Q.

Find the sum of the following series to infinity :

(i) $1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+\frac{1}{3^4}+....\infty$

(ii) $8+4\sqrt{2}+4+....\infty$

(iii) $\frac{2}{5}+\frac{3}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+.....\infty$.

(iv) 10 - 9 + 8.1 - 7.29 + ...... $\infty$

(v) $\frac{1}{3}+\frac{1}{5^2}+\frac{1}{3^3}+\frac{1}{5^4}+\frac{1}{3^5}+\frac{1}{5^6}+.....\infty$

Q. An infinite G.P. has first term $x$ and sum $5.$ Then which of the following is true

1. $x>10$
2. $x>5$
3. $0<x<10$
4. $0<x<15$

Q. Find the sum to $n$ terms of a G.P. $\sqrt{7},\sqrt{21},3\sqrt{7}...$

Q. If $\alpha ,\beta$ are the roots of the equation $\left[125\right]{\left[\begin{array}{cc}0& \frac{1}{2}\\ -1& \frac{1}{2}\end{array}\right]}^5{\left[\begin{array}{cc}1& -1\\ 2& 0\end{array}\right]}^{10}{\left[\begin{array}{cc}0& \frac{1}{2}\\ -1& \frac{1}{2}\end{array}\right]}^5\left[\begin{array}{c}{x}^2-5x+20\\ x+2\end{array}\right]=\left[40\right]$ , then $(1-\alpha )(1-\beta )-50$ is equal to ___

Q. In a G.P. of positive terms, if any term is equal to the sum of the next two terms, then the common ratio of the G.P. is

1. $2\cos {18}^{\circ }$
2. $\sin {18}^{\circ }$
3. $2\sin {18}^{\circ }$
4. $\cos {18}^{\circ }$

Q. Find the sum to $n$ terms of G.P., $x^3,x^5,x^7...$ (if $x\ne \pm 1$)

Q.

Express the recurring decimal 0.125125125... as a rational number.

Q. If $a,b,c$ are in A.P, then the determinant
$\left|\begin{array}{ccc}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ is

1. $0$
2. $1$
3. $x$
4. $2x$

Q.

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

Q.

Given that x > 0, the sum ${\sum }_{n=1}^{\infty }{\left(\frac{x}{x+1}\right)}^{n-1}$ equals

1. x

2. x + 1

3. $\frac{x}{2x+1}$

4. $\frac{x+1}{2x+1}$

Q.

Find the rotional number whose decimal expansion is $0.\overline{4}23$.

Q.

Which term of the series $\sqrt{2},\frac{2}{3},\frac{2\sqrt{2}}{9},......$... is $\frac{16}{2187}$?

1. ${10}^{th}$term

2. $8^{th}$term

3. $9^{th}$term

4. ${11}^{th}$term

Q. Find the sum to $n$ terms of a G.P., $1,-a,a^2,-a^3....$ (if $a\ne -1$)

Q.

Find the rational numbers having the following decimal expansions :

(i) $0.\overline{3}$

(ii) $0.\overline{231}$

(iii) $3.\overline{52}$

(iv) $0.6\overline{8}$

Q.

Find the geometric series whose $5^{\mathrm{th}}$ and $8^{\mathrm{th}}$ terms are $80$ and $640$, respectively.

Q. A ball is dropped from a height of $900$ centimetres. Each time it rebounds, it rises to two-third of the height it has fallen through. The total distance travelled by the ball before it comes to rest in metres, is

1. $36$
2. $45$
3. $18$
4. $27$

Q. The complex number $z\left(Re\right(z)\ne 2)$ satisfies the equation $z^2=4z+|z{|}^2-\frac{16}{|z{|}^3},$ then the value of $|z{|}^4$ is

1. $1$
2. $4$
3. $256$
4. $16$

Q.

In an infinite geometric series, the first term is $a$ and the common ratio is $r$. If the sum of the series is $4$ and the second term is $\frac{3}{4}$, then $(a,r)$ is:

1. $\left(\frac{4}{7},\frac{3}{7}\right)$

2. $\left(2,\frac{3}{8}\right)$

3. $\left(\frac{3}{2},\frac{1}{2}\right)$

4. $\left(3,\frac{1}{4}\right)$

Q.

Robin bought a computer for $\$1,250$. It will depreciate, or decrease in value, by $10\%$ each year that she owns it.

a) Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? Explain.
b) Write an explicit formula to represent the sequence.
c) Find the value of the computer at the beginning of the $6^{th}\mathrm{year}$.

Q.

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

Q. If $Z$ is an idempotent matrix, then $(I+Z{)}^n$ is equal to

1. $I+2^{n}Z$
2. $I+(2^n-1)Z$
3. $I-(2^n-1)Z$
4. None of the above

Q. Evaluate the given limit :
$\underset{x\to 4}{lim}\frac{4x+3}{x-2}$

Q. The value of xyz is $\frac{15}{2}or\frac{18}{5}$ according as the series a, x, y, z b is arithmetic or harmonic then ab =

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