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

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

$$\sum_{n=0}^{100} 3^{n}$$
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) $\frac { 11 } { 6 }$

Let x be a positive integer such that
$$\frac { 10 x } { x + 3 }$$
is equal to the square of an integer. What is the sum of the values that x can take?
A) 26
B) 27
C) 29
D) 31
E) 32

Let $(a_n)$ be a geometric sequence. The equality
$$\frac { a _ { 5 } - a _ { 1 } } { \left( a _ { 3 } \right) ^ { 2 } - \left( a _ { 1 } \right) ^ { 2 } } = \frac { 4 } { 9 }$$
is given. Given that $a _ { 2 } = \frac { 3 } { 2 }$, what is $a _ { 4 }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 27 } { 8 }$
E) $\frac { 27 } { 4 }$

For a geometric sequence $(a_n)$ with all positive terms and common ratio $r$
$$\begin{aligned} & a_1 + \frac{1}{2} + r \\ & a_7^2 = a_5 + 12 \cdot a_3 \end{aligned}$$
the equalities are given.