The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + ... is

(A) 3450                      (B) 3420                   (C) 3520                     (D) 3250

If gcd (m, n) = 1 and $1^2-2^2+3^2-4^2+...+{\left(2021\right)}^2-{\left(2022\right)}^2+{\left(2023\right)}^2=1012$$m^{2}n$, then $m^2-n^2$ is equal to

(A) 200                       (B) 180                     (C) 220                   (D) 240

If $S_n=4+11+21+34+50+...$ to $n$ terms, then $\frac{1}{60}(S_{29}-S_9)$ is equal to

(A) 223                         (B) 220                         (C) 226                       (D) 227

Let $S_1,S_2,S_3, ............ S_{10}$ respectively be the sum to 12 terms of 10 A.P. whose first terms are 1, 2, 3, ....... 10 and the common difference are 1, 3, 5, .........., 19 respectively. Then $\sum _{i=1}^{10}{S}_i$ is equal to

(A) 7220                       (B) 7380                    (C) 7260                     (D) 7360

If $a_n=\frac{-2}{4n^2-16n+15}$, then $a_1+a_2+...+a_{25}$ is equal to

(A) 52/147                    (B) 50/141                    (C) 51/144               (D) 49/138

Let $a_1,a_2,a_3,....$ 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

(A) 381/4                       (B) 9                     (C) 33/4                    (D) 24

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

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 $a_1=8,a_2,a_3,...,a_n$ 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 _________ .

The sum of the common terms of the following three arithmetic progressions.

3, 7, 11, 15, .., 399,

2, 5, 8, 11, ..., 359 and 2, 7, 12, 17, ..., 197, 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 $7^{th},9^{th},{17}^{th}$ terms of $A_1,A_2,A_3,$ respectively such that $\left|\begin{array}{ccc}a& 7& 1\\ 2b& 17& 1\\ c& 17& 1\end{array}\right|+70=0.$

If $a=29,$ then the sum of first 20 terms of an A.P. whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to ________ .

The $8^{th}$ common term of the series

$S_2=3+7+11+15+19+...,$

$S_2=1+6+11+16+21+.... .$

is _________ .

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

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to _________ .

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

(A) 231                       (B) 241                    (C) 210                      (D) 220

Let $a_1,a_2,a_3,......$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be 2 and the product of its $3^{rd}$ and $5^{th}$ terms be $\frac{1}{9}.$ Then $6(a_2+a_4)(a_4+a_6)$ is equal to

(A) $2\sqrt{2}$                        (B) 2                     (C) 3                   (D) $3\sqrt{3}$

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

(A) 6                           (B) 2                     (C) 12                        (D) - 6 

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

(A) 343                        (B) 216                      (C) $\frac{343}{8}$                    (D) $\frac{125}{8}$

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

(A) $\frac{9}{2}$                         (B) 3                       (C) 7                        (D) 14

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 $xy+yz+zx=\frac{3}{\sqrt{2}}xyz,$ then $3{(x+y+z)}^2$ is equal to _________ .

Suppose $a_1,a_2,2,a_3,a_4$ be in an arithmetic geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $\frac{49}{2}$, then $a_4$ is equal to _______ .

Let $S=109+\frac{108}{5}+\frac{107}{5^2}+...+\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $(16S-{\left(25\right)}^{-54})$ is equal to _________ .

For $k\in \mathrm{\mathbb{N}}$, if the sum of the series $1+\frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+...$ is 10, then the value of k is __________ .

The $4^{th}$ term of GP is 500 and its common ratio is $1/m,m\in N.$ Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6>S_5+1$ and $S_7<S_6+\frac{1}{2}$, then the number of possible values of $m$ 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 $16a+12b$ is equal to ________ .

Let $a_1,a_2,a_3,.....$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1{a}_9+a_2{a}_4{a}_9+a_5+a_7$ is equal to________ .

Let $\left\{a_k\right\}$ and $\left\{b_k\right\}$ $k\in \mathrm{\mathbb{N}},$ be two G.P.s with common ratios $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1<r_2.$ Let $c_k=a_k+b_k,$ $k\in \mathrm{\mathbb{N}}$. If $c_2=5$ and $c_3=\frac{13}{4}$ then $\sum _{k=1}^{\infty }{c}_k-(12a_6+8b_4)$ is equal to ________ .

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11.$ If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

(A) 55                       (B) 108                     (C) 90                     (D) 110

Let $A_1$ and $A_2$ be two arithmetic means and $G_1,G_2,G_3$ 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

(A) $2(A_1+A_2)G_1^2{G}_3^2$                      (B) ${(A_1+A_2)}^2{G}_1{G}_3$

(C) $(A_1+A_2)G_1^2{G}_3^2$                         (D) $2(A_1+A_2)G_1{G}_3$

Let $S_k=\frac{1+2+...+k}{k}$ and $\sum _{j=1}^n{S}_j^2=\frac{n}{A}(B{n}^2+Cn+D)$, where $A,B,C,D\in N$ and A has least value. Then

(A) A + B is divisible by D

(B) A + B + C + D is divisible by 5

(C) A + C + D is not divisible by B

(D) A + B = 5(D - C)

Let $a_n$ be the $n^{th}$ term of the series 5 + 8 + 14 + 23 + 35 + 50 + ... and $S_n=\sum _{k=1}^n{a}_k.$ Then $S_{30}-a_{40}$ is equal to

(A) 11280                      (B) 11290                    (C) 11310                     (D) 11260

Let $<a_n>$ be a sequence such that $a_1+a_2+...+a_n=\frac{n^2+3n}{(n+1)(n+2)}$. If $28\sum _{k=1}^{10}\frac{1}{a_k}=p_1{p}_2{p}_3...p_m,$ where $p_1,p_2,....,p_m$ are the first $m$ prime numbers, then $m$ is equal to

(A) 5                      (B) 6                             (C) 7                        (D) 8

The sum to 10 terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+...$ is

(A) $\frac{56}{111}$                   (B) $\frac{58}{111}$                   (C) $\frac{55}{111}$                    (D) $\frac{59}{111}$

The sum $\sum _{n=1}^{\infty }\frac{2n^2+3n+4}{\left(2n\right)!}$ is equal to

(A) $\frac{11e}{2}+\frac{7}{2e}-4$                                              (B) $\frac{11e}{2}+\frac{7}{2e}$

(C) $\frac{13e}{4}+\frac{5}{4e}-4$                                             (D) $\frac{13e}{4}+\frac{5}{4e}$

If ${\left(20\right)}^{19}+2\left(21\right){\left(20\right)}^{18}+3{\left(21\right)}^2{\left(20\right)}^{17}+....+20{\left(21\right)}^{19}=k{\left(20\right)}^{19},$ then $k$ is equal to _________ .

The sum to 20 terms of the series $2·2^2-3^2+2·4^2-5^2+2·6^2-..........$ is equal to _______ .

If the sum of the series $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{2^2}-\frac{1}{2·3}+\frac{1}{3^2})+(\frac{1}{2^3}-\frac{1}{2^2·3}+\frac{1}{2·3^2}-\frac{1}{3^3})+(\frac{1}{2^4}-\frac{1}{2^3·3}+\frac{1}{2^2·3^2}-\frac{1}{2·3^3}+\frac{1}{3^4})+...$

is $\frac{\alpha }{\beta },$ where $\alpha$ and $\beta$ are co-prime, then $\alpha +3\beta$ is equal to ________ .

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

Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1),b_n=b_{n-1}+a_{n-1},\forall n\ge 2.$ If $S=\sum _{n=1}^{10} \frac{b_n}{2^n}$ and $T=\sum _{n=1}^8\frac{n}{2^{n-1}},$ then $2^7(2S-T)$ is equal to _________ .

The sum $1^2-2·3^2+3·5^2-4·7^2+5·9^2-.....+15·{29}^2$ is _______.