Question 1 :
This sum to infinity of the series, $\displaystyle 1 + 2 \left ( 1 - \frac{1}{n} \right ) + 3 \left ( 1 - \frac{1}{n} \right )^2 + ...$ is<br>

Question 2 :
In a GP the ratio of the sum of the first {tex}11{/tex} terms to the sum of the last {tex}11{/tex} terms is {tex} \frac { 1 } { 8 } {/tex} and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is {tex} 2 . {/tex} Then the number of terms of the GP is less than

Question 3 :
If $1\cdot3+3\cdot3^{2}+5\cdot3^{3}+7\cdot3^{4}+\cdots$ up to $n$ terms is equal to $3+\left ( n-1 \right )3^{t}$, then $t$=<br/>

Question 4 :
In a geometric progression with common ration $q$ the sum of the first $109$ terms exceeds the sum of the first $100$ terms by $12$. If the sum of the first nine terms of the progression is $\dfrac{\lambda}{q^{100}}$, then the value of $\lambda$ equals to

Question 5 :
The sum to infinity of the series $1+\displaystyle \dfrac{2}{3}+\dfrac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+$ ... is :<br>

Question 6 :
The sum to $50$ terms of the series $1+2\left(\dfrac {1 1 }{ 50} \right) +3{ \left(\dfrac { 11 }{ 50} \right)}^{ 2 }+\dots $ is given by<br>

Question 7 :
If the sum of $n$ terms of a G.P., with common ratio r, beginning with the $pth$ terms is $k$ times the sum of an equal number of terms of the same series beginning with the $qth$ terms, then the value of $k$ is _____.

Question 8 :
The sum of first two terms of an infinite G.P. is $5$ and each term is three times the sum of following terms, then the 4th term of the series is $\_\_\_\_.$ <br/>

Question 9 :
The sum to infinity of the series $1-3x+{ 5x }^{ 2 }-{ 7x }^{ 3 }+\cdots \infty$, when $|x|&lt;1$, is<br>

Question 11 :
If the sum to infinity of the series $3+(3+d)\cfrac { 1 }{ 4 } +(3+2d)\cfrac { 1 }{ { 4 }^{ 2 } } +\cdot \cdot \cdot $ is $\cfrac { 44 }{ 9 } $, then find $d$.<br/>

Question 12 :
If the sum to infinity of the series $1+2r+{ 3r }^{ 2 }+{ 4r }^{ 3 }+\cdot \cdot \cdot $ is $\dfrac{9}{4}$, then the value of $r$ is<br/>

Question 14 :
The sum to infinity of the series $1+\displaystyle \dfrac{2}{3}+\dfrac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+$ ... is :<br>

Question 15 :
Three unequal numbers $a$, $b$, $c$ are in Arithmetic Progression and $b - a$, $c - b$, $a$ are in Geometric Progression.<br><br>The value of $\displaystyle \frac{a^{3}+b^{3}+c^{3}}{3abc}$ is