If $f ( x + y ) = f ( x ) f ( y )$ and $\sum_{x=1}^{n} f(x) = 2$, then the value of $\sum_{x=1}^{n} f(x)$ is given. [Content truncated in source]

If $\frac { 1 } { 2 \cdot 3 ^ { 10 } } + \frac { 1 } { 2 ^ { 2 } \cdot 3 ^ { 9 } } + \ldots + \frac { 1 } { 2 ^ { 10 } \cdot 3 } = \frac { K } { 2 ^ { 10 } \cdot 3 ^ { 10 } }$, then the remainder when $K$ is divided by 6 is
(1) 2
(2) 3
(3) 4
(4) 5

Let $f : N \rightarrow R$ be a function such that $f( x + y ) = 2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1) = 2$, then the value of $\alpha$ for which $\sum _ { k = 1 } ^ { 10 } f ( \alpha + k ) = \frac { 512 } { 3 } ( 2 ^ { 20 } - 1 )$ holds, is
(1) 3
(2) 4
(3) 5
(4) 6

The sum $1 + 2 \cdot 3 + 3 \cdot 3^2 + \ldots + 10 \cdot 3^9$ is equal to
(1) $\frac{2 \cdot 3^{12} + 10}{4}$
(2) $\frac{19 \cdot 3^{10} + 1}{4}$
(3) $5 \cdot 3^{10} - 2$
(4) $\frac{9 \cdot 3^{10} + 1}{2}$

Let $A_1, A_2, A_3, \ldots\ldots$ be an increasing geometric progression of positive real numbers. If $A_1 A_3 A_5 A_7 = \frac{1}{1296}$ and $A_2 + A_4 = \frac{7}{36}$, then, the value of $A_6 + A_8 + A_{10}$ is equal to
(1) 43
(2) 33
(3) 37
(4) 48

Consider two G.Ps. $2,2 ^ { 2 } , 2 ^ { 3 } , \ldots$ and $4,4 ^ { 2 } , 4 ^ { 3 } , \ldots$ of 60 and $n$ terms respectively. If the geometric mean of all the $60 + n$ terms is $( 2 ) ^ { \frac { 225 } { 8 } }$, then $\sum _ { k = 1 } ^ { n } k ( n - k )$ is equal to:
(1) 560
(2) 1540
(3) 1330
(4) 2600

If $\frac { 6 } { 3 ^ { 12 } } + \frac { 10 } { 3 ^ { 11 } } + \frac { 20 } { 3 ^ { 10 } } + \frac { 40 } { 3 ^ { 9 } } + \ldots + \frac { 10240 } { 3 } = 2 ^ { n } \cdot m$, where $m$ is odd, then $m \cdot n$ is equal to $\_\_\_\_$.

If $(20)^{19} + 2(21)(20)^{18} + 3(21)^{2}(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$, then $k$ is equal to $\_\_\_\_$.

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a G.P. of increasing positive numbers. Let the sum of its $6 ^ { \text {th} }$ and $8 ^ { \text {th} }$ terms be 2 and the product of its $3 ^ { \text {rd} }$ and $5 ^ { \text {th} }$ terms be $\frac { 1 } { 9 }$. Then $6 ( a _ { 2 } + a _ { 4 } )( a _ { 4 } + a _ { 6 } )$ is equal to
(1) 3
(2) $3 \sqrt { 3 }$
(3) 2
(4) $2 \sqrt { 2 }$

Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then $\mathbf { a } _ { 1 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 2 } \mathbf { a } _ { 4 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 5 } + \mathbf { a } _ { 7 }$ is equal to

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
(1) 7
(2) $\frac{9}{2}$
(3) 3
(4) 14

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
(1) 241
(2) 231
(3) 210
(4) 220

Let the positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ and $a _ { 5 }$ be in a G.P. Let their mean and variance be $\frac { 31 } { 10 }$ and $\frac { m } { n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac { 31 } { 10 }$ and $a _ { 3 } + a _ { 4 } + a _ { 5 } = 14$, then $m + n$ is equal to $\_\_\_\_$ .

If each term of a geometric progression $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ with $\mathrm { a } _ { 1 } = \frac { 1 } { 8 }$ and $\mathrm { a } _ { 2 } \neq \mathrm { a } _ { 1 }$, is the arithmetic mean of the next two terms and $\mathrm { S } _ { \mathrm { n } } = \mathrm { a } _ { 1 } + \mathrm { a } _ { 2 } + \ldots + \mathrm { a } _ { \mathrm { n } }$, then $\mathrm { S } _ { 20 } - \mathrm { S } _ { 18 }$ is equal to
(1) $2 ^ { 15 }$
(2) $- 2 ^ { 18 }$
(3) $2 ^ { 18 }$
(4) $- 2 ^ { 15 }$

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
(1) 7
(2) 4
(3) 5
(4) 6

Let $a$ and $b$ be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to
(1) 20
(2) 25
(3) 21
(4) 24

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th} } , 6 ^ { \text {th} }$ and $8 ^ { \text {th} }$ terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a G.P. of increasing positive terms. If $a _ { 1 } a _ { 5 } = 28$ and $a _ { 2 } + a _ { 4 } = 29$, then $a _ { 6 }$ is equal to:
(1) 628
(2) 812
(3) 526
(4) 784

Q63. In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th } } , 6 ^ { \text {th } }$ and $8 ^ { \text {th } }$ terms is equal to :
(1) 96
(2) 91
(3) 84
(4) 78

If $\mathbf{a}_{\mathbf{1}}, \mathbf{a}_{\mathbf{2}}, \mathbf{a}_{\mathbf{3}}, \ldots$ are in increasing geometric progression such that
$a_{1} + a_{3} + a_{5} = 21$,
$a_{1}a_{3}a_{5} = 64$
then $a_{1} + a_{2} + a_{3}$ is
(A) 7 (B) 10 (C) 12 (D) 15

Let product of 3 terms in G.P. is 27. If sum of these 3 terms lies in the interval $\mathbf { R } - \mathbf { ( a , b ) }$, then $\mathbf { a } ^ { \mathbf { 2 } } \boldsymbol { + } \mathbf { b } ^ { \mathbf { 2 } }$ is equal to

$S$ is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the first 3 terms of S. The $7^{\text{th}}$ term of S is 360. Find the $1^{\text{st}}$ term of S.

A geometric sequence has first term $a$ and common ratio $r$, where $a$ and $r$ are positive integers and $r$ is greater than 1.
The sum of the first $n$ terms of this sequence is denoted by $S _ { n }$
It is given that the terms of the sequence satisfy
$$S _ { 30 } - S _ { 20 } = k S _ { 10 }$$
for some positive integer $k$.
What is the smallest possible value of $k$ ?

$$\sum_{n=0}^{100} 3^{n}$$
What is the remainder when this sum is divided by 5?
A) 0
B) 1
C) 2
D) 3
E) 4

The first three terms of a geometric sequence are $\mathbf { a } + \mathbf { 3 }$, a, and $\mathbf { a } - \mathbf { 2 }$ respectively. Accordingly, what is the fourth term?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 6 }$