The series of positive multiples of 3 is divided into sets: $\{ 3 \} , \{ 6,9,12 \} , \{ 15,18,21,24,27 \} , \ldots$ Then the sum of the elements in the $11 ^ { \text {th} }$ set is equal to $\_\_\_\_$.

For three positive integers $p , q , r , x ^ { p q ^ { 2 } } = y ^ { q r } = z ^ { p ^ { 2 } r }$ and $r = p q + 1$ such that 3, $3 \log _ { y } x , 3 \log _ { z } y , 7 \log _ { x } z$ are in A.P. with common difference $\frac { 1 } { 2 }$. The $r - p - q$ is equal to
(1) 2
(2) 6
(3) 12
(4) - 6

If $a_n = \frac{-2}{4n^2 - 16n + 15}$, then $a_1 + a_2 + \ldots + a_{25}$ is equal to:
(1) $\frac{51}{144}$
(2) $\frac{49}{138}$
(3) $\frac{50}{141}$
(4) $\frac{52}{147}$

Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7 = 3$, the product $(a_1 a_4)$ is minimum and the sum of its first $n$ terms is zero then $n! - 4a_{n(n+2)}$ is equal to
(1) $\frac{381}{4}$
(2) 9
(3) $\frac{33}{4}$
(4) 24

The sum to 10 terms of the series $\frac{1}{1 + 1^2 + 1^4} + \frac{2}{1 + 2^2 + 2^4} + \frac{3}{1 + 3^2 + 3^4} + \ldots$ is:
(1) $\frac{59}{111}$
(2) $\frac{55}{111}$
(3) $\frac{56}{111}$
(4) $\frac{58}{111}$

If $\frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \ldots\ldots \text { upto } n \text { terms } } { 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots.. \text { upto } n \text { terms } } = \frac { 9 } { 5 }$ then the value of $n$ is

The sum $1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$ is $\_\_\_\_$.

Let $A _ { 1 }$ and $A _ { 2 }$ be two arithmetic means and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ be three geometric means of two distinct positive numbers. Then $G _ { 1 } ^ { 4 } + G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 } + G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$ is equal to
(1) $\left( A _ { 1 } + A _ { 2 } \right) ^ { 2 } G _ { 1 } G _ { 3 }$
(2) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } G _ { 3 }$
(3) $\left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$
(4) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$

Let $a, b, c > 1$, $a^{3}$, $b^{3}$ and $c^{3}$ be in A.P. and $\log_{a} b$, $\log_{c} a$ and $\log_{b} c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a + 4b + c}{3}$ and the common difference is $\frac{a - 8b + c}{10}$ is $-444$, then $abc$ is equal to
(1) 343
(2) 216
(3) $\frac{343}{8}$
(4) $\frac{125}{8}$

Let the digits $a , b , c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let $0 < z < y < x$ be three real numbers such that $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in an arithmetic progression and $x , \sqrt{2} y , z$ are in a geometric progression. If $x y + y z + z x = \frac { 3 } { \sqrt { 2 } } x y z$, then $3 ( x + y + z ) ^ { 2 }$ is equal to

Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A + 1 , A + 2$, respectively. Let $a , b , c$ be the $7 ^ { \text {th} } , 9 ^ { \text {th} } , 17 ^ { \text {th} }$ terms of $A _ { 1 } , A _ { 2 } , A _ { 3 }$, respectively such that $\left| \begin{array} { l l l } a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1 \end{array} \right| + 70 = 0$. If $a = 29$, then the sum of first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\frac { d } { 12 }$, is equal to $\_\_\_\_$.

The $8^{\text{th}}$ common term of the series $$\begin{aligned} & S_{1} = 3 + 7 + 11 + 15 + 19 + \ldots \\ & S_{2} = 1 + 6 + 11 + 16 + 21 + \ldots \end{aligned}$$ is

For the two positive numbers $a , b$, if $a , b$ and $\frac { 1 } { 18 }$ are in a geometric progression, while $\frac { 1 } { a } , 10$ and $\frac { 1 } { b }$ are in an arithmetic progression, then $16 a + 12 b$ is equal to $\_\_\_\_$.

Suppose f is a function satisfying $\mathrm { f } ( \mathrm { x } + \mathrm { y } ) = \mathrm { f } ( \mathrm { x } ) + \mathrm { f } ( \mathrm { y } )$ for all $\mathrm { x } , \mathrm { y } \in \mathbb { N }$ and $\mathrm { f } ( 1 ) = \frac { 1 } { 5 }$. If $\sum _ { n = 1 } ^ { m } \frac { f ( n ) } { n ( n + 1 ) ( n + 2 ) } = \frac { 1 } { 12 }$ then m is equal to $\_\_\_\_$ .

The sum of the series $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots + \frac{1}{(2n-1)(2n+1)}$ is $\_\_\_\_$.

Let $a_1 = 8, a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is $\_\_\_\_$.

Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5 = 2a_7$ and $a_{11} = 18$, then $12\left(\frac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$ is equal to $\underline{\hspace{1cm}}$.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is $\_\_\_\_$.

Let $S _ { a }$ denote the sum of first $n$ terms an arithmetic progression. If $S _ { 20 } = 790$ and $S _ { 10 } = 145$, then $S _ { 15 } - S _ { 5 }$ is
(1) 395
(2) 390
(3) 405
(4) 410

The number of common terms in the progressions $4,9,14,19 , \ldots$. up to $25 ^ { \text {th} }$ term and $3,6,9,12 , \ldots$. up to $37 ^ { \text {th} }$ term is:
(1) 9
(2) 5
(3) 7
(4) 8

In an A.P., the sixth term $\mathbf { a } _ { 6 } = 2$. If the $\mathbf { a } _ { 1 } \mathbf { a } _ { 4 } \mathbf { a } _ { 5 }$ is the greatest, then the common difference of the A.P., is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 8 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 5 } { 8 }$

The value of $\frac { 1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + \ldots + 100 \times ( 101 ) ^ { 2 } } { 1 ^ { 2 } \times 2 + 2 ^ { 2 } \times 3 + \ldots + 100 ^ { 2 } \times 101 }$ is
(1) $\frac { 32 } { 31 }$
(2) $\frac { 31 } { 30 }$
(3) $\frac { 306 } { 305 }$
(4) $\frac { 305 } { 301 }$

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15} - S_5$ is equal to:
(1) 800
(2) 890
(3) 790
(4) 690

If $\log _ { e } a , \log _ { e } b , \log _ { e } c$ are in an $A . P$. and $\log _ { e } a - \log _ { e } 2 b , \log _ { e } 2 b - \log _ { e } 3 c , \log _ { e } 3 c - \log _ { e } a$ are also in an $A . P$. , then $a : b : c$ is equal to
(1) $9 : 6 : 4$
(2) $16 : 4 : 1$
(3) $25 : 10 : 4$
(4) $6 : 3 : 2$