In an A.P., the sixth term $a_{6} = 2$. If the product\n $a_{1}a_{4}a_{5}$ is the greatest, then the common difference of the\n A.P. is equal to

The fourth power of the common difference of an arithmetic progression\n with integer entries is added to the product of any four consecutive\n terms of it, prove that the resulting sum is square of an integer.

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

Let $S_{n}$ denote the sum of first $n$ terms of an arithmetic\n progression. If and , then\n is

The real number $x_{1},x_{2},x_{3}$ satisfying the equation are in A.P. Find the intervals in which and lie.

Let $a_{1},a_{2},a_{3},\ldots$. be in an arithmetic progression of positive terms. n Let . If and , then is equal to

The sum of the common terms of the following three arithmetic\n progressions.\n 3, 7, 11, 15,......399,\\\n 2, 5, 8, 11,.........359 and $2,7,12,17,\ldots,197$, is equal to\n $\$\n\n

The area of the region $\begin{cases}(x,y):x^2≤y≤8-x^2,y≤7\end{cases}$ is.

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

The sum of all natural numbers \textquotesingle{} $n$\n \textquotesingle{} such that $100 < n < 200$ and H.C.F.\n is

If the sum of the first $n$ terms of the series\n $\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \ldots$ is\n , then equals

If $n$ arithmetic means are inserted between $a$ and 100 such that the ratio of the first mean to the last mean is $1:7$ and , then the value of is

Let $ABC$ be an equilateral triangle. A new triangle is formed by\n joining the middle points of all sides of the triangle $ABC$ and the\n same process is repeated infinitely many times. If $P$ is the sum of\n perimeters and is the sum of areas of all the triangles formed\n in this process, then

The area of the region enclosed by the parabolas $y=4x-x^2$ and $3y=(x-4)^2$ is equal to

If each term of a geometric progression $a_{1},a_{2},a_{3}$, with and , is the arithmetic mean of the next two terms and , then is equal to

If $x > 1,y > 1,z > 1$ are in G.P. then ,\n are in

Let $T_{r}$ be the $r^{\text{th}\text{~}}$ term of an A.P. for , If for some positive integers we have and , then equals

Let $f(x)$ be a polynomial function of second degree. If\n $f(1) = f( - 1)$ and are in A.P., then\n and are in

If $\alpha,\beta$ are the roots of $ax^{2} + bx + c = 0$ and\n , \n are in G.P. and , then

Let $a_1, a_2, a_3, \ldots \ldots, a 100$ be an arithmetic progression with and . For any integer with . If does not depend on .n then is .