Sequences and Series — JEE Advanced PYQs

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x)>0$ for all $x\in \mathbb{R}$, and $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$.

Let the real numbers …

Let $\mathrm{75...57}$ denote the $(\mathrm{r}+2)$ digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum $\mathrm{S}=77+757+7557+\dots +75\dots 57$. If $\mathrm{S}=\frac{75\dots 57+\mathrm{m}}{\mathrm{n}}$, where m and n are natur …

Let $l_1,l_2,\dots ,l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1,w_2,\dots ,w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1{d}_2=10$. For each …

Let $a_1,a_2,a_3,\dots$ be an arithmetic progression with $a_1=7$ and common difference 8. Let $T_1,T_2,T_3,\dots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n\ge 1$. Then, which of the following is/are TRUE ?

For positive integer n, define

Then, the value of ${\lim }_{n\to \infty }f(n)$ is equal to

Let

where $\mathrm{r}>0$. Consider the geometric progression …

Let

where $\mathrm{r}>0$. Consider the geometric progression …

Let m be the minimum possible value of ${\log }_3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y_1,y_2,y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let M be the maximum possible value of $\left({\log }_3{x}_1+{\log }_3{x}_2+{\log }_3{x}_3\right)$, where $x_1,x_2,x_3$ are positive real n …

Let $a_1,a_2,a_3,\dots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_1,b_2,b_3,\dots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_1=b_1=c$, then the number of all possible va …

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$, with $\alpha >\beta$. For all positive integers n, define

Then which o …

Let AP (a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0$. If $\mathrm{AP}(1;3)\cap \mathrm{AP}(2;5)\cap \mathrm{AP}(3;7)=\mathrm{AP}(\mathrm{a};\mathrm{d})$ then a + d equals ____

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11,\dots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23,\dots$. Then, the number of elements in the set $X\cup Y$ is ____.

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation $x^2-x-1=0$, where $\alpha \ne \beta$. For $n=0,1,2,\dots$, let $a_n=p{\alpha }^n+q{\beta }^n$.

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$. …

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation $x^2-x-1=0$, where $\alpha \ne \beta$. For $n=0,1,2,\dots$, let $a_n=p{\alpha }^n+q{\beta }^n$.

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$.

I …

The least value of $\alpha \in \mathbb{R}$ for which $4\alpha x^2+\frac{1}{x}\ge 1$, for all $x>0$, is

Let $b_i>1$ for $i=1,2,\dots ,101$. Suppose ${\log }_e{b}_1,{\log }_e{b}_2,\dots ,{\log }_e{b}_{101}$ are in Arithmetic Progression (A.P.) with the common difference ${\log }_{e}2$. Suppose $a_1,a_2,\dots ,a_{101}$ are in A.P. such that $a_1=b_1$ and …

The value of ${\sum }_{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{(k-1)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Let $a,b,c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a,b,c$ are in geometric progression and the arithmetic mean of $a,b,c$ is $b+2$, then the value of

is

Let $S_n={\sum }_{k=1}^{4n}(-1{)}^{\frac{k(k+1)}{2}}{k}^2$. Then $S_n$ can take value(s)

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k-20=$

Let $a_1,a_2,a_3,\dots$ be in harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n<0$ is

Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0. Let $b_n=$ the number of such $n$-digit integers ending with digit 1 and $c_n=$ the number of such $n$-digit integers ending with d …

Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0. Let $b_n=$ the number of such $n$-digit integers ending with digit 1 and $c_n=$ the number of such $n$-digit integers ending with d …

The value of $6+{\log }_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\dots }}}\right)$ is

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$, with $\alpha >\beta$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is

Let $a,b$ and $c$ be three real numbers satisfying

Let $b=6$, with $a$ and …

The minimum value of the sum of real numbers $a^{-5},a^{-4},3a^{-3},1,a^8$ and $a^{10}$ with $a>0$ is

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{11}$ be real numbers satisfying …

Let ${\mathrm{S}}_{\mathrm{k}},\mathrm{k}=1,2,\dots ,100$, denote the sum of the infinite geometric series whose first term is $\frac{\mathrm{k}-1}{\mathrm{k}!}$ and the common ratio is $\frac{1}{\mathrm{k}}$. Then the value of …

If the sum of first $n$ terms of an A.P. is $c{n}^2$, then the sum of squares of these $n$ terms is

Suppose four distinct positive numbers $a_1,a_2,a_3,a_4$ are in G.P. Let $b_1=a_1$, $b_2=b_1+a_2$, $b_3=b_2+a_3$ and $b_4=b_3+a_4$.

STATEMENT-1 : The numbers $b_1,b_2,b_3,b_4$ are neither in A.P. nor in G.P.
and
STATEMENT-2 …

Let $V_r$ denote the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2r-1)$. Let


The sum …

Let $V_r$ denote the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2r-1)$. Let


$T_r$ is always

Let $V_r$ denote the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2r-1)$. Let


Which one of the following …

Let $A_1,G_1,H_1$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n\ge 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n,G_n,H_n$ respectively.

Which one of t …

If ${\mathrm{a}}_{\mathrm{n}}=\frac{3}{4}-{\left(\frac{3}{4}\right)}^2+{\left(\frac{3}{4}\right)}^3+\cdots (-1{)}^{\mathrm{n}-1}{\left(\frac{3}{4}\right)}^{\mathrm{n}}$ and ${\mathrm{b}}_{\mathrm{n}}=1-{\mathrm{a}}_{\mathrm{n}}$, then find the minimum natural number ${\mathrm{n}}_0$ s …