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

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

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

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

If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ are in A.P. such that $a _ { 1 } + a _ { 7 } + a _ { 16 } = 40$, then the sum of the first 15 terms of this A.P. is

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is
(1) 1356
(2) 1365
(3) 1256
(4) 1465

If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \ldots$. is $( 102 ) \mathrm { m }$, then m is equal to
(1) 20
(2) 25
(3) 5
(4) 10

If the $10^{\text{th}}$ term of an A.P. is $\frac{1}{20}$, and its $20^{\text{th}}$ term is $\frac{1}{10}$, then the sum of its first 200 terms is.
(1) 50
(2) $50\frac{1}{4}$
(3) 100
(4) $100\frac{1}{2}$

If the sum of first 11 terms of an A.P. , $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\ldots$ is $0 \left( a _ { 1 } \neq 0 \right)$ then the sum of the A.P $a _ { 1 } , a _ { 3 } , a _ { 5 } , \ldots\ldots a _ { 23 }$ is $k a _ { 1 }$ where $k$ is equal to
(1) $- \frac { 121 } { 10 }$
(2) $\frac { 121 } { 10 }$
(3) $\frac { 72 } { 5 }$
(4) $- \frac { 72 } { 5 }$

Let $S _ { 1 }$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S _ { 2 }$ be the sum of first $4 n$ terms of the same arithmetic progression. If ( $S _ { 2 } - S _ { 1 }$ ) is 1000 , then the sum of the first $6 n$ terms of the arithmetic progression is equal to:
(1) 1000
(2) 7000
(3) 5000
(4) 3000

Let $S _ { n }$ denote the sum of first $n$-terms of an arithmetic progression. If $S _ { 10 } = 530 , S _ { 5 } = 140$, then $S _ { 20 } - S _ { 6 }$ is equal to:
(1) 1862
(2) 1842
(3) 1852
(4) 1872

Let $a _ { 1 } , \quad a _ { 2 } , \quad a _ { 3 } , \quad \ldots$ be an A.P. If $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 10 } } { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } = \frac { 100 } { p ^ { 2 } }$, then $\frac{a_{11}}{a_{10}}$ is equal to:

The remainder on dividing $1 + 3 + 3 ^ { 2 } + 3 ^ { 3 } + \ldots + 3 ^ { 2021 }$ by 50 is $\_\_\_\_$.

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 $\_\_\_\_$.

Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { \mathrm { n } } , \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and $110 < a _ { 15 } < 120$, then the sum of the first ten terms of the progression is equal to
(1) 290
(2) 380
(3) 460
(4) 510

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5 . Let the sum of its first five terms be $\frac { 98 } { 25 }$. Then the sum of the first 21 terms of an AP, whose first term is $10 a r , n ^ { \text {th } }$ term is $a _ { n }$ and the common difference is $10 a r ^ { 2 }$, is equal to
(1) $21 a _ { 11 }$
(2) $22 a _ { 11 }$
(3) $15 a _ { 16 }$
(4) $14 a _ { 16 }$

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 $\_\_\_\_$.

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) $\frac{125}{8}$

The sum of all those terms, of the arithmetic progression $3, 8, 13, \ldots, 373$, which are not divisible by 3, is equal to $\_\_\_\_$.

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

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

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:
(1) 150
(2) 180
(3) 160
(4) 125

An arithmetic progression is written in the following way

			2
	11	5	14	8	17
20		23		26		29

The sum of all the terms of the $10 ^ { \text {th} }$ row is $\_\_\_\_$

Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is
(1) 20
(2) 90
(3) 45
(4) 25

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
(1) - 1080
(2) - 1020
(3) - 1200
(4) - 120