Sequences and Series

An arithmetic progression is written in the following way The sum of all the terms of the $10^{\text {th }}$ row is_______

Let the positive integers be written in the form : If the $k^{\text {th }}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is _______

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

Q4899 (image question)

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…

Let $A_{1},A_{2},A_{3},\ldots\,..$ be squares such that for each $n⩾1,$ the length of the side of $A_{n}$ equals the length of diagonal of $A_{n+1}$. If the length of $A_{1}$ is $12\mathrm{cm}$, then the smallest value…

Let $3,7,11,15,..,403$ and $2,5,8,11,...,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to_________

Let$3,a,b,c$ be in $A.P.$ and$3,a-1,b+1,c+9$ be in $G.P.$ Then, the arithmetic mean of $a,b$and $c$is:

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a, b)$, then $a^{2}+b^{2}$ is equal to $\_\_\_\_$.

If $P11+2\cdot\,P22+3\cdot\,P33+\ldots\,+15\cdot\,P1515=Prq-s,0\leq\,s\leq\,1$, then $Cr-sq+s$ is equal to

Let $S_{n}$ be the sum to n-terms of an arithmetic progression $3,7,11,\ldots\,\ldots\,$, if $40<\left(\frac{6}{n(n+1)}\underset{k=1}{\overset{n}{\sum\,}}S_{k}\right)<42$, then $n$ equals ____________.

If in a G.P. of $64$ terms, the sum of all the terms is $7$times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to