If the sum of infinite G.P. $p, 1, \dfrac{1}{p}, \dfrac{1}{p^2}, ......., $ is $\dfrac{9}{2}$. Then find the value of $p$.

The sum of 100 terms of the series .9+.09+.009.....will be:

Sum $1,\sqrt { 3 } ,3......$ to $12$ terms is

The sum of $2n$ terms of a geometric progression whose first term is $'a'$ and common ratio $'r'$ is equal to the sum of $n$ terms of a geometric progression whose first term is $'b'$ and common '$r^{2}$'. then $b$ is equal to

The value of $x$ that satisfies the relation $x=1-x+x^{2}-x^{3}+x^{4}-x^{5}+.\infty$ if $|x|<1$ 

In a infinite G.P. , the sum of first three terms is 70. If the extreme terms are multiplied by 4 and the middle term is multiplied by 5, the resulting terms form an A.p. then the sum to infinite terms of G.p.   

Find the sum of $1,\dfrac 14,\dfrac 1{16},.....$

If a $ >0,  $ then the minimum value of sum of $  \dfrac{1}{a}, 1, a^{2}, a^{3}, \dfrac{1}{a^{4}}  $ is equal to

Evaluate:
$2+2^2+2^3+....+2^9=$

Sum of the first five terms of the geometric series $1 + \dfrac {2}{3} + \dfrac {4}{9} + $....is 

Given $A=2^{65}$ and $B=(2^{64}+2^{63}+2^{62}+....+2^0)$

The sum of the geometric sequence is given as $S=\cfrac{a(1-r^n)}{1-r}$, where $r$ is the

If $a _1,\, a _2,\, a _3,\dots,a _n$ are in geometric progression. Then the given geometric progression is a

$x, 2x, 4x, . . .$
The first term in the sequence above is $x$, and each term thereafter is equal to twice the previous term. Find the sum of the first five terms of this sequence.

Find the sum of the following G.P. to $n$ terms $0.5 + 0.55 + 0.555 + 0.5555 + .....$

Let $n > 1$ be the positive integer. The largest positive integer $m$, such that $n^m + 1$ divides $1 + n + n^2 ..... n^{125}$ is

The sum of the first three terms of an increasing G.P. is $13$ and their product is $27$. The sum of the first $5$ terms is,

If $i^{2}=-1$, then sum $i+i^{2}+i^{3}+.......$ to $1000$ terms is equal to

The sum of sequence $0.15,0.015,0.0015,.....$ upto 20 term is ?

The sum of $10$ terms of GP $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \ldots$ is-

The geometric series $a+ar+ar^{2}+ar^{3}+......\infty$ has sum $7$ and the terms involving odd powders of $r$ has sum $'3'$, then the value of $(a^{2}-r^{2})$ is-

The sum 
$1 + \left( {1 + x} \right) + \left( {1 + x + {x^2}} \right) + \left( {1 + x + {x^2} + {x^3}} \right) +  \ldots n$  terms equals 

$\lim _{ x\leftarrow 1 }{ \cfrac { x+{ x }^{ 2 }+{ x }^{ 3 }+....+{ x }^{ n }-n }{ x-1 }  } =$

The sum of first $10$ terms of the series $\sqrt{2}+\sqrt{6}+\sqrt{18}+...$ is

If ${S} _{n}=\sum _{ r=1 }^{ n }{ \cfrac { 1+2+{ 2 }^{ 2 }+..Sum\quad to\quad r\quad terms }{ { 2 }^{ r } }  } $, then ${S} _{n}$ is equal to 

Find the sum of 8 terms of the G.P: 3+6+12+24.........

If $a _{0},a _{1},a _{3},....$ and $b _{0},b _{1},b _{2},b _{3},...$ are two geometric progressions with $a _{1}=2\surd 3$ and $b _{1}=\dfrac {52}{9}\sqrt {3}$ if $3a _{99}b _{99}=104$ then $\displaystyle \sum^{101} _{i=0}a _{1}b _{1}$ is

$1+3+7+15+31+.....$ to n terms 

If $1+a+a^{2}+a^{3}+.........+a^{n}=(1+a)(1+a^2)(1+a^4)$ then $n$ is given by 

The sum of first 4 term of GP with $a=2,r=3$ is 

Three numbers whose sum is $45$ are in A.P. If $5$ is subtracted from the first number and $25$ is added to third number, the numbers are in G.P. Then numbers can be

If $S$ is the sum to infinity of a $G.P.$ whose first terms is $a$ then the sum of the first $n$ terms is 

For first $n$ natural numbers we have the following results with usual notations $ \displaystyle \sum _{r=1}^{n}r =\frac{n(n+1)}{2}, \sum _{r=1}^{n}r^{2} =\frac{n(n+1)(2n+1)}{6},\sum _{r=1}^{n}r^{3}=\left ( \sum _{r=1}^{n}r \right )^{2}$ If $\displaystyle a _{1}a _{2}....a _{n} \in A.P $ then sum to $n$ terms of the sequence $\displaystyle \frac{1}{a _{1}a _{2}},\frac{1}{a _{2}a _{3}},...\frac{1}{a _{n-1}a _{n}}$ is equal to $\displaystyle \frac{n-1}{a _{1}a _{n}}$
 and the sum to $ n$ terms of a $G.P$ with first term '$a$' & common ratio '$r$' is given by  $\displaystyle S _{n}= \frac{lr-a}{r-1}$ for $ r \neq 1 $ for $ r =1 $ sum to $n$ terms of same $G.P.$ is $n$ $a$, where the sum to infinite terms of$G.P.$ is the limiting value of
 $\displaystyle \frac{lr-a}{r-1} $ when $\displaystyle n \rightarrow \infty ,\left |  r \right | < l $ where $l$ is the last term of $G.P.$  On the basis of above data answer the following questionsThe sum of the series $\displaystyle 2+6+18+...+486 $ equals?

$\displaystyle x(x+y)+x^{2}(x^{2}+y^{2})+x^{3}(x^{3}+y^{3})+$.......to n terms.

Find the value of the sum $\displaystyle \sum _{r=1}^{n}\,$ $\displaystyle \sum _{s=1}^{n}\, \delta _{rs}\, 2^r\, 3^s$ where $ \delta _{rs}$ is zero if $r \neq s$ & $\delta _{rs}$ is one if $r=s$

The A.M. of the series $1, 2, 4, 8, 16, ......, 2$$^n$ is

$\displaystyle 1+\frac{1}{4\times 3}+\frac{1}{4\times 3^{2}}+\frac{1}{4\times 3^{3}}$  is equal to 

$6^{1/2}\, .\, 6^{1/4}\, .\, 6^{1/8}\, ..... \infty\, =\, ?$ 

In a geometric progression with common ratio '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 $\displaystyle \frac {\lambda}{q^{100}}$ then the value of $ \lambda $ equals to

Let $\displaystyle S=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ find the sum of first $20$ terms of the series

The $n^{th}$ term of the sequence 

$\displaystyle\frac{1}{100}$, $\displaystyle\frac{1}{10000}$, $\displaystyle\frac{1}{1000000}$, $\dots\dots$ is

Find $S _n$, the sum of the first $n$ terms, for the following geometric series. $a _1=120, a _5= 1, r=-2$.

Find the sum of the first $6$ terms of the geometric series $80 - 20 + 5 +.....$

Find the sum of the geometric series $4 + 2 + 1 +... +$ $\dfrac{1}{16}$

What is $S _6$ of the geometric progression $6, 12, 24...$?

Find $3 + 12 + 48 +...$ up to $5$ terms.

Determine the sum of the first 8 terms of the G.P. $1, 2, 4, 8...$

Evaluate the sum of the first nine terms of the geometric sequence $5, 10, 20,...$

Calculate the sum of first $20$ terms of the G.P. $-1, 1, -1, 1....$

The sum of $6^{th}$ term in the geometric series $4, 12, 36...$ is

What is the sum of G.P. $1, 3, 9, 27,.....$ up to $7$ numbers?

What is the sum of the first five terms of the geometric sequence $5, 15, 45, ... $?

Find $4 + 12 + 36 +..... $ upto $6$ terms.

If the first term and common ratio is $5$, find the sum of 8th term of infinite G.P.

The value of $1 + 2 + 4 + 8....$ of G.P., where $n=6$  is

Calculate sum of eleventh term of the geometric sequence $3, 6, 12, 24, ... $

The sum of first $n$ terms of an G.P. is

A rubber ball is dropped from a height of $10$ meters. If the ball always rebounds $\dfrac {4}{5}$ the distance it has fallen, calculate, how far, in meters, will the ball have travelled at the moment it hits the ground for the fourth time?

Find the sum of odd integers between $1$ and $1000$ which are divisible by $3$.

How many terms of the series $1+3+9+ ...$sum to $121$?

What is the sum of first eight terms of the series $1-\cfrac { 1 }{ 2 } +\cfrac { 1 }{ 4 } -\cfrac { 1 }{ 8 } +.....$?

What is the greatest value of the positive integer n satisfying the condition $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} +  ...... + \dfrac{1}{2^{n - 1}} < 2 - \dfrac{1}{1000}$?

The value of the sum $\sum _{ n=1 }^{ 13 }{ \left( { i }^{ n }+{ i }^{ n+1 } \right)  } $ where $i=\sqrt { -1 } $ is:

Sum $1 + 2a + 3a^{2} + 4a^{3} + ....$ to $n$ terms.

The geometric mean if the series $1, 2, 4,...., 2^n$, is

If the sum $1+2+3 +....+ K$ is a perfect square N$^{2}$ and if N is less than 100, then the possible values for K are: 

The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\dfrac {1}{2}$. The first term of the progression is

The sum of $2n$ terms of a series of which every even term is $'a'$ times the terms before it, and every odd term $'c'$ times the terms before it, the first term being unity, is

The sum of $10$ terms of the series $0.7 + .77 + .777 + \ldots \ldots \ldots$ is

The sum of series $\displaystyle \frac{3}{4} + \frac{15}{16} + \frac{63}{64}+ ..... $ up to $n$ terms is

If the sum of $n$ terms of a GP (with common ratio $r$) beginning with the $\displaystyle p^{th}$ term is $k$ times the sum of an equal number of the same series beginning with the $\displaystyle q^{th}$ term, then the value of $k$ is

The sum of $1 + \dfrac {2}{5} + \dfrac {3}{5^{2}} + \dfrac {4}{5^{3}} + ....$ up to $n$ terms is

In a $G.P$. the ratio of the sum of the first eleven terms to the sum of last eleven terms is $\displaystyle \frac{1}{8}$ and the ratio of the sum of all terms without the first nine to the sum of all the terms without the last nine is $2$. Then the number of terms of the $G.P$ is