The common difference of the A.P. $b_{1},b_{2},\ldots,b_{m}$ is 2 more than common difference of A.P. $\mathrm{a}_{1},\mathrm{a}_{2},\ldots,\mathrm{a}_{\mathrm{n}}$. If $\mathrm{a}_{40}=-159,\mathrm{a}_{100}=-399$ and $\mathrm{b}_{100}=\mathrm{a}_{70}$, then $\mathrm{b}_{1}$ is equal to:
(1) 81
(2) $-127$
(3) $-81$
(4) 127

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac { 1 } { 2 }$, then the greatest number amongst them is
(1) 27
(2) 7
(3) $\frac { 21 } { 2 }$
(4) 16

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 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) $- \frac { 72 } { 5 }$

If $1 + \left( 1 - 2 ^ { 2 } \cdot 1 \right) + \left( 1 - 4 ^ { 2 } \cdot 3 \right) + \left( 1 - 6 ^ { 2 } \cdot 5 \right) + \ldots \ldots + \left( 1 - 20 ^ { 2 } \cdot 19 \right) = \alpha - 220 \beta$, then an ordered pair $( \alpha , \beta )$ is equal to:
(1) $( 10,97 )$
(2) $( 11,103 )$
(3) $( 10,103 )$
(4) $( 11,97 )$

If $3 ^ { 2 \sin 2 \alpha - 1 } , 14$ and $3 ^ { 4 - 2 \sin 2 \alpha }$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is
(1) 66
(2) 81
(3) 65
(4) 78

Let $a , b , c$ be in arithmetic progression. Let the centroid of the triangle with vertices $( a , c ) , ( 2 , b )$ and $( a , b )$ be $\left( \frac { 10 } { 3 } , \frac { 7 } { 3 } \right)$. If $\alpha , \beta$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, then the value of $\alpha ^ { 2 } + \beta ^ { 2 } - \alpha \beta$ is:
(1) $- \frac { 71 } { 256 }$
(2) $\frac { 69 } { 256 }$
(3) $\frac { 71 } { 256 }$
(4) $- \frac { 69 } { 256 }$

The sum of 10 terms of the series $\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$ is :
(1) $\frac { 143 } { 144 }$
(2) $\frac { 99 } { 100 }$
(3) 1
(4) $\frac { 120 } { 121 }$

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

If $\alpha , \beta$ are natural numbers such that $100 ^ { \alpha } - 199 \beta = ( 100 ) ( 100 ) + ( 99 ) ( 101 ) + ( 98 ) ( 102 ) + \ldots . + ( 1 ) ( 199 )$, then the slope of the line passing through $( \alpha , \beta )$ and origin is:
(1) 540
(2) 550
(3) 530
(4) 510

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 21 }$ be an A.P. such that $\sum _ { n = 1 } ^ { 20 } \frac { 1 } { a _ { n } a _ { n + 1 } } = \frac { 4 } { 9 }$. If the sum of this A.P. is 189 , then $\mathrm { a } _ { 6 } \mathrm { a } _ { 16 }$ is equal to :
(1) 57
(2) 48
(3) 36
(4) 72

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:

A function $f ( x )$ is given by $f ( x ) = \frac { 5 ^ { x } } { 5 ^ { x } + 5 }$, then the sum of the series $f \left( \frac { 1 } { 20 } \right) + f \left( \frac { 2 } { 20 } \right) + f \left( \frac { 3 } { 20 } \right) + \ldots + f \left( \frac { 39 } { 20 } \right)$ is equal to:
(1) $\frac { 19 } { 2 }$
(2) $\frac { 49 } { 2 }$
(3) $\frac { 39 } { 2 }$
(4) $\frac { 29 } { 2 }$

If $x = \sum _ { n = 0 } ^ { \infty } a ^ { n } , y = \sum _ { n = 0 } ^ { \infty } b ^ { n } , z = \sum _ { n = 0 } ^ { \infty } c ^ { n }$, where $a , b , c$ are in A.P. and $| a | < 1 , | b | < 1 , | c | < 1 , a b c \neq 0$, then
(1) $x , y , z$ are in A.P.
(2) $x , y , z$ are in G.P.
(3) $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in A.P.
(4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1 - ( a + b + c )$

If $\frac { 1 } { ( 20 - a ) ( 40 - a ) } + \frac { 1 } { ( 40 - a ) ( 60 - a ) } + \ldots\ldots + \frac { 1 } { ( 180 - a ) ( 200 - a ) } = \frac { 1 } { 256 }$, then the maximum value of $a$ is
(1) 198
(2) 202
(3) 212
(4) 218

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

If $\left\{ a _ { i } \right\} _ { i = 1 } ^ { \mathrm { n } }$, where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum _ { i = 1 } ^ { n } a _ { i } = 192 , \sum _ { i = 1 } ^ { \frac { n } { 2 } } a _ { 2 i } = 120$, then $n$ is equal to
(1) 18
(2) 36
(3) 96
(4) 48

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) $14 a _ { 16 }$

If $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots$ and $b _ { 1 } , b _ { 2 } , b _ { 3 } \ldots$ are A.P. and $a _ { 1 } = 2 , a _ { 10 } = 3 , a _ { 1 } b _ { 1 } = 1 = a _ { 10 } b _ { 10 }$ then $a _ { 4 } b _ { 4 }$ is equal to
(1) $\frac { 28 } { 27 }$
(2) $\frac { 28 } { 24 }$
(3) $\frac { 23 } { 26 }$
(4) $\frac { 22 } { 23 }$

If $p$ and $q$ are real numbers such that $p + q = 3 , p ^ { 4 } + q ^ { 4 } = 369$, then the value of $\left( \frac { 1 } { p } + \frac { 1 } { q } \right) ^ { - 2 }$ is equal to (if the full expression were available).

Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^ { 2 } - 8ax + 2a = 0$ and $q$ and $s$ are the roots of the equation $x ^ { 2 } + 12bx + 6b = 0$, such that $\frac { 1 } { p }, \frac { 1 } { q }, \frac { 1 } { r }, \frac { 1 } { s }$ are in A.P., then $a ^ { - 1 } - b ^ { - 1 }$ is equal to $\_\_\_\_$.

$\frac { 2 ^ { 3 } - 1 ^ { 3 } } { 1 \times 7 } + \frac { 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 2 \times 11 } + \frac { 6 ^ { 3 } - 5 ^ { 3 } + 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 3 \times 15 } + \ldots \ldots + \frac { 30 ^ { 3 } - 29 ^ { 3 } + 28 ^ { 3 } - 27 ^ { 3 } + \ldots + 2 ^ { 3 } - 1 ^ { 3 } } { 15 \times 63 }$ is equal to $\_\_\_\_$ .

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

Let $a _ { 1 } = b _ { 1 } = 1$, $a _ { n } = a _ { n - 1 } + 2$ and $b _ { n } = a _ { n } + b _ { n - 1 }$ for every natural number $n \geq 2$. Then $\sum _ { n = 1 } ^ { 15 } a _ { n } \cdot b _ { n }$ is equal to $\_\_\_\_$.