Basics of Sequence and Series

Let $x\in R$. Find $\sum _{K=0}^{\infty }\left[\frac{x+2^K}{2^{K+1}}\right]$.

Consider the sequence $a_1,a_2,a_3,\dots$ such that $a_1=1,a_2=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_n$ for $n=1,2,3,\dots$. If $\left(\frac{a_1+\frac{1}{a_2}}{a_3}\right)·\left(\frac{a_2+\frac{1}{a_3}}{a_4}\right)·\left(\frac{a_3+\frac{1}{a_4}}{a_5}\right)\dots .\left(\frac{a_{30}+\frac{1}{a_{31}}}{a_{32}}\right)=2^{\alpha }\left({}^{61}C_{31}\right)$, then $\alpha$ is

If ${\tan }^{-1}\left(\frac{12}{31}\right)+{\tan }^{-1}\left(\frac{12}{139}\right)+\dots \dots +{\tan }^{-1}\left(\frac{12}{36n^2-5}\right)={\tan }^{-1}\left(\frac{\mathrm{An}}{\mathrm{Bn}+13}\right)$. Then which is/are correct $\left(A,B\in N\right)$

In a sequence of six numbers, every term after the second term is equal to sum of the previous two terms. Also the last term is four times the first term, and the sum of all six terms is $13$. What is the first term?

In a sequence $\left\{a_n\right\},a_1=2,a_{n+1}=1-\frac{1}{a_n}$ for $n\ge 1,n\in N$. Let $P_n$ be the product of its first $n$ terms, then the value of $P_{2023}$ is equal to

Define difference between sequences and progressions. Find the first five terms of arithmetic progression with $n^{th}$ term $T_n=4n+3$.

Define difference between sequences and progressions. Find the first five terms of arithmetic progression with $n^{th}$ term $T_n=4n-3$.

Define difference between sequences and progressions. Find the first five terms of arithmetic progression with $n^{th}$ term $T_n=2n$.

Define difference between sequences and progressions. Find the first five terms of arithmetic progression with $n^{th}$ term $T_n=2n+1$.

Define a real sequence and find the first five terms of sequence $T_n=4n+3$.

Define a real sequence and find the first five terms of sequence $T_n=4n-3$.

Define a real sequence and find the first five terms of sequence $T_n=2n$.

Define a real sequence and find the first five terms of sequence $T_n=2n+1$.

Define difference between sequences and progressions. Find the first five terms of arithmetic progression with $n^{th}$ term $T_n=2n-1$.

Define a real sequence and find the first five terms of sequence $T_n=2n-1$.

Write the next three terms of following progression.

$6,12,18,24,\dots$

Consider the sum $S=\sum _{i=3}^x(3i+1)$, where $x$ is a positive integer greater than $3$.

Write down the first three terms of the series.

Write the next term of the sequence $0, 2, 6, 12, 20, \dots$.

First term of a sequence is $1$ and the $(n+1{)}^{th}$ term is obtained by adding $(n+1)$ to the $n^{th}$ term for all natural numbers $n$. Find the sixth term of the sequence.

If the $n^{\mathrm{th}}$ term of the sequence $2, 5, 10, 17, \dots \dots$ is of the form $a{n}^2+b$, then find the value of $a+b$.