Basics of Sequence and Series

Let $\left[\alpha \right]$ denote the greatest integer $\le \alpha$. Then $\left[\sqrt{1}\right]+\left[\sqrt{2}\right]+\left[\sqrt{3}\right]+.............+\left[\sqrt{120}\right]$ 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$\_\_\_\_\_\_\_\_\_\_.$

Let $s_1,s_2,s_3....,s_{10}$ respectively be the sum of $12$ terms of $10 \mathrm{A}.\mathrm{Ps}$ whose first terms are $1, 2, 3,....,10$ and the common differences are $1, 3, 5,...,19$ respectively. Then $\sum _{i=1}^{10}{s}_i$ is equal to

Let $<a_n>$ be a sequence such that $a_1+a_2+...+a_n=\frac{n^2+3n}{\left(n+1\right)\left(n+2\right)}$. 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

Let $S_K=\frac{1+2+...+K}{K}$ and $\sum _{j=1}^n{S^2}_j=\frac{n}{A}\left(B{n}^2+Cn+D\right)$ where $A, B, C, D \in N$ and $A$ Has least value then

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

Evaluate:

$S=\sum _{n=0}^{\infty }\frac{2^n}{\left(a^{2^n}+1\right)}$ (where $a>1$)

If $t_n$ denotes the $n^{\mathrm{th} }$ term of the sequence $1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,\dots \dots$, then $\frac{t_{255}}{t_{257}}$ is equal to

The series $1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\dots .$ is 

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 $a+b$.

Write the multiple of $6$ out of first $5$ terms of the following sequence whose $n^{th}$ term is given by:

$a_n=n\left(n+2\right)$

Write the multiple of $5$ out of first $5$ terms of the following sequence whose $n^{th}$ term is given by:

$a_n=n(n-1)$

Write the sum of first five terms of the following sequence:
$a_1=1, a_n=a_{n-1}+2, n>1$.

Find the sum of first four terms of the sequence defined by $a_1=3$ and, $a_n=3a_{n-1}+2$, for all $n>1$.

A sequence is given by $a_n=-10n+3$.

 Find the the smallest value of $k$ for which $a_k<-1000$.

Consider the sequence $u_n=3+4\times (n-1)$. Find $u_6$. 

In a triangle $ABC$ if $\frac{\sin A-\sin C}{\cos C-\cos A}=\cot B$, then $A, B, C$ are in

The first three terms in the sequence $a_n=2n-7, \forall n\in N$ are 

The first three terms in the sequence $a_n=2n+7, \forall n\in N$ are 

If $S_n=nP+\frac{n\left(n-1\right)}{2}Q$, where $S_n$ denotes the sum of the first $n$ terms of an $\mathrm{A}.\mathrm{P}.$, then the common difference is