Let $7 \overbrace { 5 \cdots 5 } ^ { r } 7$ denote the $( r + 2 )$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7 \overbrace { 5 \cdots 5 } ^ { 98 } 7$. If $S = \frac { 7 \overbrace { 5 \cdots 5 } ^ { 99 } 7 + m } { n }$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
(1) 19 months
(2) 20 months
(3) 21 months
(4) 18 months

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right)p^{2} - 2p(ab+bc+cd) + \left(b^{2}+c^{2}+d^{2}\right) \leq 0$, then
(1) $a, b, c, d$ are in A.P.
(2) $ab = cd$
(3) $ac = bd$
(4) $a, b, c, d$ are in G.P.

If the A.M. between $p ^ { \text {th} }$ and $q ^ { \text {th} }$ terms of an A.P. is equal to the A.M. between $r ^ { \text {th} }$ and $s ^ { \text {th} }$ terms of the same A.P., then $p + q$ is equal to
(1) $r + s - 1$
(2) $r + s - 2$
(3) $r + s + 1$
(4) $r + s$

If $100$ times the $100^{\text{th}}$ term of an AP with non-zero common difference equals the $50$ times its $50^{\text{th}}$ term, then the $150^{\text{th}}$ term of this AP is
(1) $-150$
(2) 150 times its $50^{\text{th}}$ term
(3) 150
(4) zero

If $x, y, z$ are in AP and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in AP, then
(1) $x = y = z$
(2) $2x = 3y = 6z$
(3) $6x = 3y = 2z$
(4) $6x = 4y = 3z$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an A.P, such that $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } { a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { q } } = \frac { p ^ { 3 } } { q ^ { 3 } } ; p \neq q$. Then $\frac { a _ { 6 } } { a _ { 21 } }$ is equal to:
(1) $\frac { 41 } { 11 }$
(2) $\frac { 31 } { 121 }$
(3) $\frac { 11 } { 41 }$
(4) $\frac { 121 } { 1861 }$

If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n } , \ldots$ are in A.P. such that $a _ { 4 } - a _ { 7 } + a _ { 10 } = m$, then the sum of first 13 terms of this A.P., is :
(1) 10 m
(2) 12 m
(3) 13 m
(4) 15 m

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) $\frac { 2 \sqrt { 17 } } { 9 }$

If $( 10 ) ^ { 9 } + 2 ( 11 ) ^ { 1 } ( 10 ) ^ { 8 } + 3 ( 11 ) ^ { 2 } ( 10 ) ^ { 7 } + \ldots\ldots + 10 ( 11 ) ^ { 9 } = k ( 10 ) ^ { 9 }$, then $k$ is equal to:
(1) 100
(2) 110
(3) $\frac { 121 } { 10 }$
(4) $\frac { 441 } { 100 }$

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

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

If the sum $\frac { 3 } { 1 ^ { 2 } } + \frac { 5 } { 1 ^ { 2 } + 2 ^ { 2 } } + \frac { 7 } { 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } } + \ldots + $ up to 20 terms is equal to $\frac { k } { 21 }$, then $k$ is equal to
(1) 240
(2) 120
(3) 60
(4) 180

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) $\frac{7}{4}$

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

For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then
(1) $b,\ c$ and $a$ are in G.P.
(2) $b,\ c$ and $a$ are in A.P.
(3) $a,\ b$ and $c$ are in A.P.
(4) $a,\ b$ and $c$ are in G.P.

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 $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255