Sequences and Series

Consider the closed interval $[0,1]$ and label it as $C_0$. In the next step, divide the interval $[0,1]$ into three parts $\left[0, \frac{1}{3}\right],\left(\frac{1}{3}, \frac{2}{3}\right),\left[\frac{2}{3}, 1\right]$…

Let $N$ be the sum of all integers greater than 1000 and less than 2000 . If $\frac{N}{x} \in \mathbb{N}$, then $x$ is

Given a prime $p$ and an integer $n$, let $\nu(n)$ denote the largest integer $k$ such that $p^k$ divides $n$. Define $f(n)=p^{-\nu(n)}$. Then

Let $a_1, a_2, \ldots, a_{2022}$ be positive integers such that $a_i+a_{i+1}=b+1$ for $1 \leq i \leq 2021$, where $b$ is a constant. If $a_{202}=2$, then $a_{22}$ is

Let $d_1, d_2, d_3, \cdots, d_r$ be all the positive divisors of $n$. If $d_1+d_2+\cdots+d_r=1000$, then $\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_r}$ equals

The number of positive integers $n$ such that $2^n+7^n$ is a perfect square is

Let $\left\{a_n \mid n \in \mathbb{N}\right\}$ and $\left\{b_n \mid n \in \mathbb{N}\right\}$ be two sets of real numbers, where $\mathbb{N}$ denotes the set of natural numbers. Define $s_k=\sum_{i=1}^k a_i$ and…

Suppose $p, q, r, s$ are distinct integers chosen from $\{1,2,3,4\}$. Let $x=\frac{1}{p+\frac{1}{q+\frac{1}{r+\frac{1}{s}}}}$ The value of $s$ for which $x$ takes maximum value is

For a finite sequence $a=\left(a_1, \cdots, a_n\right)$ of $n$ integers (where $n \geq 2$), define $S(a)=\sum_{\substack{i, j \in\{1,2, \ldots, n\} \\ i \lt j}} a_i a_j .$

If the roots of the equation $x^3+b x^2+c x-1=0$ are real, positive, distinct and in geometric progression then

For $n \in \mathbb{N}$, let $f(n)= \begin{cases}3 n+1 & \text { if } n \text { is odd } \\ \frac{n}{2} & \text { if } n \text { is even }\end{cases}$ Define $f^{(1)}(n)=f(n)$ and $f^{(i)}(n)=f\left(f^{(i-1)}(n)\right)$…

Let $s(n)$ be the sum of digits of a positive integer $n$. Then the number of solutions of the equation $n+s(n)+s(s(n))=2019$ with $n \geq 2000$ is