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

Area of the region $\begin{cases}(x,y):x^2+(y-2)^2≤4,x^2≥2y\end{cases}$ is

Suppose $a,b,c$ are in A.P. and $a^{2},b^{2},c^{2}$ are in G.P. If and , then the value of is

If $\left\{ a_{i} \right\}_{i = 1}^{n}$, where is an even\n integer, is an arithmetic progression with common difference 1 , and\n ,\n and then is equal to

The coefficientof $x^{101}$ in the expression $(5 + x)^{500} + x(5 + x)^{499} + x^{2}(5 + x)^{498} + \ldots + x^{500},x > 0$ is

Let $T_{r}$ be the $r^{\text{th}\text{~}}$ term of an A.P. whose\n first term is and common difference is . If for some\n positive integers , and\n , then n equals equals

Let the first term $a$ and the common ratio $r$ of a geometric\n progression be positive integers. If the sum of squares of its first\n three terms is 33033 , then the sum of these three terms is equal to

If $19^{\text{th}\text{~}}$ term of a non-zero A.P. is zero, then if\n ( $49^{\text{th}\text{~}}$ term) :\n term is

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to :

If $A = \sum_{n = 1}^{\infty} \frac{1}{(3 + (-1)^n)^n}$ and then is equal to:

Sum of the first $n$ terms of the series n $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \ldots\ldots$.\n is equal to

For three positive integers $p,q,r,x^{pq^{2}} = y^{qr} = z^{p^{2}r}$ and such that are in A.P. with common difference . The is equal to

Let $l_{1},l_{2},\ldots,l_{100}$ be consecutive terms of and arithmetic progression with common difference , and let , be consecutive terms of another arithmetic progression with common difference , where . For each , let be a rectangle with length , width and area . If , then the value of is .

In a triangle $ABC$ if $cotA,cotB,cotC$ are in A.P., then\n are in progression.

Let $\alpha$ and $\beta$ be the roots of the equation\n $x^{2} + qx$ , where . If and \n be the consecutive terms of a non constant G.P. and\n , then the value\n of is

Let $a_{1},a_{2},\ldots,a_{n}$ be a given A.P. whose common\n difference is an integer and\n . If\n and , then the ordered\n pair is equal to

The sides of a triangle inscribed in a given circle subtend angles\n $\alpha,\beta$ and $\gamma$ at the centre. The minimum value of\n the arithmetic mean of\n $cos\left( \alpha + \frac{\pi}{2} \right),cos\left( \beta + \frac{\pi}{2} \right)$\n and is equal\n to..............

If 100 times the $100^{\text{th}\text{~}}$ term of an A.P. with\n non-zero common difference equals the 50 times its\n $50^{\text{th}\text{~}}$ term, then the \n term of this A.P. is

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

Find three numbers $a,b,c$ between 2 and 18 such that (i) their sum\n is 25 (ii) $2,a,b$ are consecutive terms of an A.P. (iii) the\n numbers are consecutive terms of a G.P.