Sum of Infinite Terms of a GP

Q.

Show that the following four conditions are equivalent:

(i) $A\subset B$ (ii) $A-B=\Phi$

(ii) $A\cup B=B$ (iv) $A\cap B=A$

Q. Find the sum to n terms of the sequence, 8, 88, 888, 8888…

Q. $Thedomainofthefunctionf\left(x\right)={[9^x+{27}^{\frac{2}{3}(x-2)}-219-3^{2(x-1)}]}^{\frac{1}{4}}$

1. 
2. 
3. [-3, 3]
4. [0, 1]

Q. If the first term of a G.P., $a_1,a_2,a_3,\dots$ is unity, then the value of $4a_2+5a_3$ will be minimum when the common ratio is

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

Q. If each term of an infinite G.P. is twice the sum of the terms following it, then the common ratio of G.P. is

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

Q.

Sum the series $\mathrm{1.2.3}+2.3.4.+3.4.5+\cdots$ to n terms.

Q. A student is constructing a family tree. If he wants to track back through $10$ generations to calculate the total number of ancestors he has, then the total number of ancestors after the ${10}^{th}$ generation is

1. $2046$
2. $1024$
3. $2048$
4. $1023$

Q.

If x, y, and z are in G.P. and $x+3,y+3,Z+3$ are in H.P., then

1. y =2

2. y =3

3. y =1

4. y=0

Q.

The sum of two numbers is $\frac{13}{6}$.

An even number of AMs are being inserted between them. The sum of means inserted exceeds the number of means by one. Find the number of AMs inserted.

Or

Find the sum of infinite series. $\frac{1}{3}+\frac{1}{5^2}+\frac{1}{3^2}+\frac{1}{5^4}+\frac{1}{3^3}+\frac{1}{5^6}+.......$

Q. In a non-zero G.P., if $T_{p-1}+T_{p+1}=3T_p,$ where $T_n$ denotes the $n^{th}$ term of the G.P., then the common ratio of the G.P. can be

1. $\frac{1+\sqrt{5}}{2}$
2. $\frac{1-\sqrt{5}}{2}$
3. $\frac{5-\sqrt{3}}{2}$
4. $\frac{3-\sqrt{5}}{2}$

Q.

If AM between two numbers exceeds their GM by $\frac{5}{2}$ and the GM exceeds their HM by 2, the ratio of numbers is

1. 1:4

2. 4:1

3. 1:2

4. 2:1

Q. Express the given complex number in the form $a+ib:$$i^9+i^{19}$

Q.

If f is a function satisfying $f(x+y)=f\left(x\right)f\left(y\right)$ for all, $x,yϵN$ such that f(1) = 3 and ${\sum }_{x=1}^{n}f\left(x\right)=120$, find the value of n.

Q. Given the matrices $A=\left[\begin{array}{cc}3& 2\\ 4& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right].$ Then sum of the absolute values of the entries of the matrices $X$ and $Y$ satisfying $AX=B$ and $YA=B$ is

Q. Express the given complex number in the form $a+ib:$${(1-i)}^4$

Q. Express the given complex number in the form $a+ib:$$i^{-39}$

Q.

The $(m+1{)}^{th}$ term of $(\frac{x}{y}+\frac{y}{x}{)}^{2m+1}$ depends on -

1. Independent of x

2. Constant

3. Independent of y

4. Depends on x, y and m

Q.

Find the real numbers x and y if $(x-iy)(3+5i)$ is the conjugate of -6 -24i

Q. Solve $\underset{x\to -2}{lim}\frac{(\frac{1}{x}+\frac{1}{2})}{x+2}$

Q. If $S_1,S_2,S_3\cdots S_p$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ... p and whose common
ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots \frac{1}{p+1}respectively,$
Then $S_1+S_2+S_3+\cdots +S_p=$

1. p
2. $\frac{p(p+3)}{2}$
3. None of these
4. $\frac{p(p+1)}{2}$

Q.

8+88+888+........=

1. None of these

2. $\frac{80}{81}({10}^n-1)-\frac{8n}{9}$

3. $\frac{10}{81}({10}^n-1)$

4. $\frac{80}{81}({10}^n-1)+\frac{8n}{9}$

Q. Find the mean and variance for the data

$X_i$	6	10	14	18	24	28	30
$f_i$	2	4	7	12	8	4	3

Q. If $f$ is a function satisfying $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y$ $\in N$ such that $f\left(1\right)=3$ and ${\sum }_{x=1}^{n}f\left(x\right)=120$, find the value of $n$ .

Q. If the sum of the series $1+\frac{2}{x}+\frac{4}{x^2}+\frac{8}{x^3}+.....\infty$ is a finite number, then the complete solution of x will be -
[UPSEAT 2002]

1. x > 2
2. x < -2
3. (x < -2) U (x > 2)
4. None of these

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

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

Q. If $x=1+a+a^2+...\infty ,(|a|<1),y=1+b+b^2+...\infty ,(|b|<1),$ then $z=1+ab+a^2{b}^2+a^3{b}^3+...\infty$ is

1. x+y-1xy
2. 2xyx+y-1
3. xyx+y-1
4. x+y-12xy

Q. For the function
$f\left(x\right)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\cdot \cdot \cdot +\frac{x^2}{2}+x+1$
Prove that $f^{{}^{\prime }}\left(1\right)=100f^{{}^{\prime }}\left(0\right)$

Q. Let $f\left(2\right)=4,f^{\prime }\left(2\right)=4$, then $\underset{x\to 2}{lim}\frac{xf\left(2\right)-2f\left(x\right)}{x-2}$ is

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

Q.

How do you calculate ${\log }_{10}\left(2\right)$?

Q. Express the given complex number in the form $a+ib:$$\left(5i\right)(-\frac{3}{5}i)$