Sequences and Series

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $\left(a,c\right),\left(2,b\right)$ and $\left(a,b\right)$ be $\left(\frac{10}{3},\frac{7}{3}\right).$ If $\alpha\,,\beta\,$ are…

$1^3-2^3+3^3-4^3+\ldots .+9^3=$

Let $f:R\rightarrow\,R$ be such that for all $x\in\,R\left(2^{1+x}+2^{1-x}\right),f\left(x\right)$ and $\left(3^{x}+3^{-x}\right)$ are in A.P., then the minimum value of $f\left(x\right)$ is

If $\sum_{r=1}^{25}\left(\frac{r}{r^{4}+r^{2}+1}\right)=\frac{p}{q}$, where $p$ and $q$ are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$。

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log…

Let $a_{n}$ be the $n^{th}$ term of a G.P.of positive terms. If $\underset{n=1}{\overset{100}{\sum\,}}a_{2n+1}=200$ and $\underset{n=1}{\overset{100}{\sum\,}}a_{2n}=100,$ then…

The number of terms common to the two A.P.'s $3,7,11,\ldots\,,407$ and $2,9,16,\ldots\,,709$ is ____________?

The sum $1+\frac{1^{3}+2^{3}}{1+2}+\frac{1^{3}+2^{3}+3^{3}}{1+2+3}+...+\frac{1^{3}+2^{3}+3^{3}+...+15^{3}}{1+2+3+...+15}-\frac{1}{2}\left(1+2+3+...+15\right)$ is equal to

If$\left|x\right|<1,\left|y\right|<1$and$x\neq\,1$, then the sum to infinity of the following series$\left(x+y\right)+\left(x^{2}+xy+y^{2}\right)+\left(x^{3}+x^{2}y+xy^{2}+y^{3}\right)+.....$is

Let the sum of the first $n$ terms of a non-constant $A.P.,a_{1},a_{2},a_{3},....,a_{n}$ be $50n+\frac{n(n-7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.$, then the ordered pair…

Let $a,1a_{2},\ldots\,,a_{n}$be a given A.P. whose common difference is an integer and $S_{n}=a_{1}+a_{2}+\ldots\,+a_{n}$. If $a_{1}=1,a_{n}=300$and $15\leq\,n\leq\,50,$ then the ordered pair…

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$more identical balls are added to the total number of…