Sequences and Series

$\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots \frac{1}{(3 n-1)(3 n+2)}=$

Let$S_{n}$ denote the sum of the first n terms of an arithmetic progression. If$S_{10}=390$and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15}-S_{5}$is equal to:

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then n is equal to

Let $a_{1},a_{2},a_{3},\ldots\,$. be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then $a_{1}a_{9}+a_{2}a_{4}a_{9}+a_{5}+a_{7}$is…

The 4th term of GP is $500$ and its common ratio is$\frac{1}{m},m\in\,N$. Let$S_{n}$denote the sum of the first $n$terms of this GP. If$S_{6}>S_{5}+1$and$S_{7}<S_{6}+\frac{1}{2}$,then the number of possible values of…

For the two positive numbers $a,b$, if $a,b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a},10$ and $\frac{1}{b}$ are in an arithmetic progression, then, $16a+12b$ is equal to _____ .

If the first term of an$A.P.$is$3$and the sum of its first$25$terms is equal to the sum of its next$15$terms, then the common difference of this$A.P.$is

Let $a_{1},a_{2},\ldots\,,a_{10}$ be an $A.P.$ with common difference $-3$ and $b_{1},b_{2},\ldots\,,b_{10}$ be a $G.P.$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k},k=1,2,\ldots\,,10.$ If $c_{2}=12$ and…

Let $S_{n}$ denote the sum of the first $n$ terms of an $A.P.$. If $S_{4}=16$ and $S_{6}=-48$ , then $S_{10}$ is equal to:

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms, the three terms now form an $A.P.$,then the sum of the original three terms of the given…

The sum$\underset{n=1}{\overset{21}{\sum\,}}\frac{3}{\left(4n-1\right)\left(4n+3\right)}$is equal to

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