For $p,q \in R$, consider the real valued function\n $f(x) = (x - p)^{2} - q,x \in R$ and $q > 0$. Let\n $a_{1},a_{2},a_{3}$ and $a_{4}$ be in an arithmetic progression\n with mean $p$ and positive common difference. If\n $\left| f\left( a_{i} \right) \right| = 500$ for all $i = 1,2,3$,\n 4 , then the absolute difference between the roots of $f(x) = 0$ is

If total number of runs scored in $n$ matches is\n $\left( \frac{n + 1}{4} \right)$\\\n where , and the runs\n scored in the match are given by\n , where . Find

Let $A = \left\{ 1,a_{1},a_{2},\ldots,a_{18},77 \right\}$ be a set of integers with . Let the set : contain exactly 39 elements. Then, the value of is equal to

The sum of the first $n$ terms of the series\\\n $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + \ldots\ldots\ldots$\n is , when is even. when is odd,\n the sum is

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

The minimum value of the sum of real numbers $a^{- 5},a^{- 4},3a^{- 3},1,a^{8}$ and with is

If $a_{1},a_{2},a_{3}\ldots$ and are in A.P., and , , then is equal to

For any odd integer $n \geq 1,n^{3} - (n - 1)^{3} + \ldots\ldots$\\\n

\begin{aligned}\n&\text { If } a_1, a_2, \ldots a_n \text { are in A.P. where } a_i>0 \text { for all } i \text {, show that }\\\n&\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\n&=\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}\n\n\end{aligned}\n

If $3^{2sin2\alpha - 1},14$ and $3^{4 - 2sin2\alpha}$ are the first three terms of an A.P. for some , then the sixth term of this A.P. is

Let a relation 𝑅 on 𝑁 × 𝑁 be defined as: (𝑥1, 𝑦1)𝑅(𝑥2, 𝑦2) if and only if\n𝑥1 ≤ 𝑥2 or 𝑦1 ≤ 𝑦2. Consider the two statements :\n(I) 𝑅 is reflexive but not symmetric.\n(II) 𝑅 is transitive\nThen which one of the following is true?\n\n

Suppose $a_{1},a_{2},\ldots,a_{n},\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is and , then the sum of the first ten terms of the progression is equal to

If $\log_{3}2,\log_{3}\left( 2^{x} - 5 \right),\log_{3}\left( 2^{x} - \frac{7}{2} \right)$ are in an arithmetic progression, then the value of is equal to

Let $S_{1}$ be the sum of first $2n$ terms of an arithmetic progression. Let be the sum of first terms of the same arithmetic progression. If is 1000 , n then the sum of the first terms of the arithmetic progression is equal to

For $x \geq 0$, the least value of $K$, for which $4^{1 + x} +$ are three consecutive terms of an A.P., is equal to

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line , is

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

If three positive numbers $a,b$ and $c$ are in A.P. such that $abc = 8$, then the minimum possible value of is

Let $a, a r, a r^2, \ldots \ldots b e$ an infinite G.P. If and , then r is equal to :

Let $a,b,c > 1,a^{3},b^{3}$ and be in A.P., and and be in G.P. If the sum of first 20 terms of an A.P., whose first term is and the common difference is is -444 , then is equal to