Sequences and Series — JEE Main PYQs

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is:

If the sum of the first 10 terms of the series $\frac{4.1}{1+4.1^4}+\frac{4.2}{1+4.2^4}+\frac{4.3}{1+4.3^4}+\dots$ is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $m+n$ is equal to ______.

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\dots$ upto $\infty$ terms, is equal to

Let $a_1,a_2,a_3,\dots$ be in an A.P. such that ${\sum }_{k=1}^{12}{a}_{2k-1}=-\frac{72}{5}{a}_1$, $a_1\ne 0$. If ${\sum }_{k=1}^n{a}_k=0$, then $n$ is:

Let $P_n={\alpha }^n+{\beta }^n,n\in \mathbf{N}$. If $P_{10}=123,P_9=76$, ${\mathrm{P}}_8=47$ and ${\mathrm{P}}_1=1$, then the quadratic equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is :

The sum $1+3+11+25+45+71+\dots$ upto 20 terms, is equal to

Let $a_1,a_2,a_3,\dots$ be a G. P. of increasing positive numbers. If ${\mathrm{a}}_3{\mathrm{a}}_5=729$ and ${\mathrm{a}}_2+{\mathrm{a}}_4=\frac{111}{4}$, then $24\left(a_1+a_2+a_3\right)$ is equal to

The sum of the infinite series ${\cot }^{-1}\left(\frac{7}{4}\right)+{\cot }^{-1}\left(\frac{19}{4}\right)+{\cot }^{-1}\left(\frac{39}{4}\right)+{\cot }^{-1}\left(\frac{67}{4}\right)+\dots$ is :-

If the sum of the first 20 terms of the series

is $\frac{\mathrm{m}}{\mathrm{n}}$, where m and n are coprime, then $\mathrm{m}+\mathrm{n}$ is equal to …

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of AP's in A and B respectiv …

$1+3+5^2+7+9^2+\dots$ upto 40 terms is equal to

Let $a_n$ be the $n^{\text{th }}$ term of an A.P.

If ${\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_1+{\mathrm{a}}_2+{\mathrm{a}}_3+\dots .+{\mathrm{a}}_{\mathrm{n}}=700,{\mathrm{a}}_6=7$ and ${\mathrm{S}}_7=7$, then ${\mathrm{a}}_{\mathrm{n}}$ is equal to :

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :

Let $x_1,x_2,x_3,x_4$ be in a geometric progression. If $2,7,9,5$ are subtracted respectively from $x_1,x_2,x_3,x_4$ then the resulting numbers are in an arithmetic progression. Then the value of $\frac{1}{24}(x_1{x}_2{x}_3{x}_4)$ is:

If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\dots \infty =\frac{{\pi }^4}{90}$,
$\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\dots =\alpha$,
$\frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\dots \infty =\beta$,
then $\frac{\alpha }{\beta }$ is equal to

Let the first three terms 2, p and $q\ne 2$, of a G.P. be respectively the $7^{th}$, $8^{th}$ and $13^{th}$ terms of an A.P. If the $5^{th}$ term of the G.P. is the $n^{th}$ term of the A.P., then $n$ is equal to

The value of $\frac{1\times 2^2+2\times 3^2+\dots +100\times (101{)}^2}{1^2\times 2+2^2\times 3+\dots +100^2\times 101}$ is

Let three real numbers $a,b,c$ be in arithmetic progression and $a+1,b,c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a,b,c$ is 8, then the cube of the geometric mean of $a,b,c$ is

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1.2}+\frac{1}{2.3}+\dots +\frac{1}{99.100}=n$, then the point $(m,n)$ lies on the line

Let $a_1,a_2,a_3,\dots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^2-a_2^2+a_3^2-a_4^2+\dots +a_{2k-1}^2-a_{2k}^2$. If $A_3=-153,A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$, then $a_{17}-A_7$ is equal to ____.

For $x\ge 0$, the least value of $K$, for which $4^{1+x}+4^{1-x},\frac{K}{2},16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to :

If $1+\frac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}+\frac{5-2\sqrt{6}}{18}+\frac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}+\frac{49-20\sqrt{6}}{180}+\dots$ upto $\infty =2+\left(\sqrt{\frac{b}{a}}+1\right){\log }_e\left(\frac{a}{b}\right)$, where $a$ and $b$ are integers with …

Let the first term of a series be $T_1=6$ and its $r^{th}$ term $T_r=3T_{r-1}+6^r$, $r=2,3,\dots ,n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}(n^2-12n+39)(4\cdot 6^n-5\cdot 3^n+1)$, then $n$ is equal to ____.

A software company sets up m number of computer system 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 v …

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is the sum of area of all the triangles formed in this pr …

$S(x)=(1+x)+2(1+x{)}^2+3(1+x{)}^3+3(1+x{)}^4+\dots +60(1+x{)}^{60},x\ne 0$

and $(60{)}^{2}S(60)=a(b{)}^b+b,a,b\in \mathbb{N}$ then $(a+b)$ is equal to ______.

Let the positive integers be written in the form :

If the ${\mathrm{k}}^{\text{th }}$ row contains exactly k numbers for every natural number k , then the row in which the number 5310 will be, is ______ .

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{\text{th }}$, $6^{\text{th }}$ and $8^{\text{th }}$ terms is:-

An arithmetic progression is written in the following way

\begin{tabular}{ccccccc}
& & & 2 & & & \\
& & 5 & & 8 & & \\
20 & 11 & & 14 & & 17 & \\
23 & & 26 & 26 & & 29
\end{tabular}

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

If the sum of series
$\frac{1}{1\cdot (1+d)}+\frac{1}{(1+d)(1+2d)}+\dots \dots +\frac{1}{(1+9d)(1+10d)}$
is equal to 5, then 50d is equal to :

If a function $f$ satisfies $f(m+n)=f(m)+f(n)$ for all $m,n\in \mathbb{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that ${\sum }_{k=1}^{2022}f(\lambda +k)\le (2022{)}^2$ is equal to

Let $\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2,\dots \dots$. be an infinite G.P. If ${\sum }_{n=0}^{\infty }{\mathrm{ar}}^n=57$ and ${\sum }_{n=0}^{\infty }{\mathrm{a}}^3{\mathrm{r}}^{3n}=9747$, then $\mathrm{a}+18\mathrm{r}$ is equal to :

Let $\alpha ,\beta ;\alpha >\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let $P_n={\alpha }^n-{\beta }^n,n\in N$. Then $(11\sqrt{3}-10\sqrt{2}){\mathrm{P}}_{10}+(11\sqrt{2}+10){\mathrm{P}}_{11}-11{\mathrm{P}}_{12}$ is equal to :

If $\left(\frac{1}{\alpha +1}+\frac{1}{\alpha +2}+\dots +\frac{1}{\alpha +1012}\right)$
$-\left(\frac{1}{2\cdot 1}+\frac{1}{4\cdot 3}+\frac{1}{6\cdot 5}+\dots +\frac{1}{2024\cdot 2023}\right)$
$=\frac{1}{2024}$, then $\alpha$ is equal to ____.

If ${\log }_e\mathrm{a},{\log }_e\mathrm{b},{\log }_e\mathrm{c}$ are in an A.P. and ${\log }_e\mathrm{a}-{\log }_{e}2b,{\log }_{e}2b-{\log }_{e}3c,{\log }_{e}3c-{\log }_{e}a$ are also in an A.P, then $\mathrm{a}:\mathrm{b}:\mathrm{c}$ is equal to

If each term of a geometric progression ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$ with ${\mathrm{a}}_1=\frac{1}{8}$ and ${\mathrm{a}}_2\ne {\mathrm{a}}_1$, is the arithmetic mean of the next two terms and …

Let $2^{\text{nd}}$, $8^{\text{th}}$ and $44^{\text{th}}$, terms of a non-constant A.P. be respectively the $1^{\text{st}}$, $2^{\text{nd}}$ and $3^{\text{rd}}$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-

Let ${\mathrm{S}}_n$ denote the sum of first n terms of an arithmetic progression. If ${\mathrm{S}}_{20}=790$ and ${\mathrm{S}}_{10}=145$, then ${\mathrm{S}}_{15}-{\mathrm{S}}_5$ is :

Let $\alpha =1^2+4^2+8^2+13^2+19^2+26^2+\dots$ upto 10 terms and $\beta ={\sum }_{n=1}^{10}{n}^4$. If $4\alpha -\beta =55k+40$, then $k$ is equal to ______.

The $20^{\text{th }}$ term from the end of the progression $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\dots ,-129\frac{1}{4}$ is :-

If $\alpha ,\beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023{\alpha }^n+2024{\beta }^n$, then

The number of common terms in the progressions $4,9,14,19$, ______, up to $25^{\text{th}}$ term and $3,6,9,12$, ......., up to $37^{\text{th}}$ term is :

If

then the value of p is ____.

Let ${\mathrm{S}}_{\mathrm{n}}$ denote the sum of the first n terms of an arithmetic progression. If ${\mathrm{S}}_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$, then ${\mathrm{S}}_{15}-{\mathrm{S}}_5$ is equal to:

If three successive terms of a G.P. with common ratio $\mathrm{r}(\mathrm{r}>1)$ are the lengths of the sides of a triangle and $[\mathrm{r}]$ denotes the greatest integer less than or equal to r , then $3[\mathrm{r}]+[-\mathrm{r}]$ is equal to :

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

Let $3,\mathrm{a},\mathrm{b},\mathrm{c}$ be in A.P. and $3,\mathrm{a}-1,\mathrm{b}+1,\mathrm{c}+9$ be in G.P. Then, the arithmetic mean of $\mathrm{a},\mathrm{b}$ and c is :

Let $3,7,11,15,\dots .,403$ and $2,5,8,11,\dots ,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ____.

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

Let a and b be be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term of another GP, whose first term is a and fifth term is b . Then p is equal to