The sum of the series: $1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \ldots\ldots$. upto 10 terms, is:
(1) $\frac { 18 } { 11 }$
(2) $\frac { 22 } { 13 }$
(3) $\frac { 20 } { 11 }$
(4) $\frac { 16 } { 9 }$

The value of $1^{2} + 3^{2} + 5^{2} + \cdots + 25^{2}$ is:
(1) 2925
(2) 1469
(3) 1728
(4) 1456

If $x, y, z$ are positive numbers in A.P. and $\tan^{-1}x, \tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then which of the following is correct.
(1) $6x = 3y = 2z$
(2) $6x = 4y = 3z$
(3) $x = y = z$
(4) $2x = 3y = 6z$

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots\ldots$, is:
(1) $\frac{7}{81}\left(179 + 10^{-20}\right)$
(2) $\frac{7}{9}\left(99 + 10^{-20}\right)$
(3) $\frac{7}{81}\left(179 - 10^{-20}\right)$
(4) $\frac{7}{9}\left(99 - 10^{-20}\right)$

Let $\alpha$ and $\beta$ be the roots of equation $p x ^ { 2 } + q x + r = 0 , p \neq 0$. If $p , q , r$ are in A.P. and $\frac { 1 } { \alpha } + \frac { 1 } { \beta } = 4$, then the value of $| \alpha - \beta |$ is
(1) $\frac { \sqrt { 34 } } { 9 }$
(2) $\frac { 2 \sqrt { 13 } } { 9 }$
(3) $\frac { \sqrt { 61 } } { 9 }$
(4) $\frac { 2 \sqrt { 17 } } { 9 }$

Let $f ( n ) = \left[ \frac { 1 } { 3 } + \frac { 3 n } { 100 } \right] n$, where $[ n ]$ denotes the greatest integer less than or equal to $n$. Then $\sum _ { n = 1 } ^ { 56 } f ( n )$ is equal to
(1) 56
(2) 1287
(3) 1399
(4) 689

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is:
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $\sqrt { 2 } + \sqrt { 3 }$
(4) $3 + \sqrt { 2 }$

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its $4^{\text{th}}$ term is:
(1) 8
(2) 24
(3) 20
(4) 16

The number of terms in an $A.P$. is even, the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac { 1 } { 2 }$, then the number of terms in the $A.P$. is
(1) 4
(2) 8
(3) 16
(4) 12

The sum of first 9 terms of the series $\frac { 1 ^ { 3 } } { 1 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } } { 1 + 3 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } } { 1 + 3 + 5 } + \ldots$ is
(1) 192
(2) 71
(3) 96
(4) 142

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac{1}{2}$, then the number of terms in the A.P. is:
(1) 4
(2) 8
(3) 12
(4) 16

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:
(1) $\frac{8}{5}$
(2) $\frac{4}{3}$
(3) $1$
(4) $\frac{7}{4}$

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is: (1) $\frac{8}{5}$ (2) $\frac{4}{3}$ (3) $1$ (4) $\frac{7}{4}$

If the sum of the first ten terms of the series $\left(1\frac{3}{5}\right)^{2}+\left(2\frac{2}{5}\right)^{2}+\left(3\frac{1}{5}\right)^{2}+4^{2}+\left(4\frac{4}{5}\right)^{2}+\ldots$, is $\frac{16}{5} m$, then $m$ is equal to: (1) 102 (2) 101 (3) 100 (4) 99

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots a _ { n } , \ldots$, be in A.P. If $a _ { 3 } + a _ { 7 } + a _ { 11 } + a _ { 15 } = 72$, then the sum of its first 17 terms is equal to :
(1) 306
(2) 204
(3) 153
(4) 612

The sum $\sum _ { r = 1 } ^ { 10 } \left( r ^ { 2 } + 1 \right) \times ( r ! )$, is equal to:
(1) $11 \times ( 11 ! )$
(2) $10 \times ( 11 ! )$
(3) $(11)!$
(4) $101 \times ( 10 ! )$

If the arithmetic mean of two numbers $a$ and $b , a > b > 0$, is five times their geometric mean, then $\frac { a + b } { a - b }$ is equal to:
(1) $\frac { 7 \sqrt { 3 } } { 12 }$
(2) $\frac { 3 \sqrt { 2 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 2 }$
(4) $\frac { 5 \sqrt { 6 } } { 12 }$

If the sum of the first $n$ terms of the series $\sqrt { 3 } + \sqrt { 75 } + \sqrt { 243 } + \sqrt { 507 } + \ldots$ is $435 \sqrt { 3 }$, then $n$ equals:
(1) 13
(2) 15
(3) 29
(4) 18

Let $a , b , c \in \mathbb { R }$. If $f ( x ) = a x ^ { 2 } + b x + c$ is such that $a + b + c = 3$ and $f ( x + y ) = f ( x ) + f ( y ) + x y$, $\forall x , y \in \mathbb { R }$, then $\sum _ { n = 1 } ^ { 10 } f ( n )$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255

Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + 2 \cdot 6 ^ { 2 } + \ldots$ If $B - 2 A = 100 \lambda$, then $\lambda$ is equal to :
(1) 496
(2) 232
(3) 248
(4) 464

Let $\frac { 1 } { x _ { 1 } } , \frac { 1 } { x _ { 2 } } , \ldots , \frac { 1 } { x _ { n } } \left( x _ { i } \neq 0 \right.$ for $\left. i = 1,2 , \ldots , n \right)$ be in A.P. such that $x _ { 1 } = 4$ and $x _ { 21 } = 20$. If $n$ is the least positive integer for which $x _ { n } > 50$, then $\sum _ { i = 1 } ^ { n } \left( \frac { 1 } { x _ { i } } \right)$ is equal to
(1) 3
(2) $\frac { 1 } { 8 }$
(3) $\frac { 13 } { 4 }$
(4) $\frac { 13 } { 8 }$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots , a _ { 49 }$ be in $A.P$. such that $\sum _ { k = 0 } ^ { 12 } a _ { 4 k + 1 } = 416$ and $a _ { 9 } + a _ { 43 } = 66$. If $a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 17 } ^ { 2 } = 140 m$, then $m$ is equal to:
(1) 33
(2) 66
(3) 68
(4) 34

If $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h ^ { 2 } } , \ldots \ldots \frac { 1 } { h _ { n } }$ are two A.P's such that $x _ { 3 } = h _ { 2 } = 8$ and $x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 }$. $h _ { 10 }$ equals.
(1) 2560
(2) 2650
(3) 3200
(4) 1600

If $x _ { 1 } , x _ { 2 } , \ldots\ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h _ { 2 } } , \ldots\ldots , \frac { 1 } { h _ { n } }$ are two A.P.s such that $x _ { 3 } = h _ { 2 } = 8 \& x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 } \cdot h _ { 10 }$ is equal to
(1) 3200
(2) 1600
(3) 2650
(4) 2560

If three distinct numbers $a , b , c$ are in G.P. and the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?
(1) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in A.P.
(2) $d , e , f$ are in A.P.
(3) $d , e , f$ are in G.P.
(4) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in G.P.