Sequences and Series

$a_{12} =$

If, for a positive integer n, the quadratic equation,
$$x(x+1)+(x+1)(x+2)+\dots+(x+\overline{n-1})(x+n)=10n$$
has two consecutive integral solutions, then n is equal to :

For any three positive real numbers a, b and c,
$$9(25a^2+b^2)+25(c^2-3ac)=15b(3a+c).$$
Then :

Let a, b, c ∈ ℝ. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy$, ∀ x, y ∈ ℝ, then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to :

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then $\dfrac{a+b}{a-b}$ is equal to :

If the sum of the first n terms of the series $\sqrt3+\sqrt{75}+\sqrt{243}+\sqrt{507}+\dots$ is $435\sqrt3$, then n equals :

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

Let $S_n=\dfrac1{1^3}+\dfrac{1+2}{1^3+2^3}+\dfrac{1+2+3}{1^3+2^3+3^3}+\dots+\dfrac{1+2+\dots+n}{1^3+2^3+\dots+n^3}$. If $100S_n=n$, then n is equal to :

If $\displaystyle\lim_{n\to\infty}\frac{1^a+2^a+\dots+n^a}{(n+1)^{a-1}[(na+1)+(na+2)+\dots+(na+n)]}=\frac1{60}$ for some positive real number a, then a is equal to :

The least value of $\alpha\in\mathbb R$ for which $4\alpha x^2+\dfrac1x\ge1$, for all $x>0$, is

Let $b_i>1$ for $i=1,2,\ldots,101$. Suppose $\log_eb_1,\log_eb_2,\ldots,\log_eb_{101}$ are in Arithmetic Progression (A.P.) with the common difference $\log_e2$. Suppose $a_1,a_2,\ldots,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\cdots+b_{51}$ and $s=a_1+a_2+\cdots+a_{51}$, then

The value of $\displaystyle\sum_{k=1}^{13}\frac{1}{\sin\left(\frac{\pi}{4}+\frac{(k-1)\pi}{6}\right)\sin\left(\frac{\pi}{4}+\frac{k\pi}{6}\right)}$ is equal to

If the 2$^{\text{nd}}$, 5$^{\text{th}}$ and 9$^{\text{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

If the sum of the first ten terms of the series $\left(1\dfrac35\right)^2+\left(2\dfrac25\right)^2+\left(3\dfrac15\right)^2+4^2+\left(4\dfrac45\right)^2+\dots$, is $\dfrac{16}5m$, then m is equal to :

Let x, y, z be positive real numbers such that x + y + z = 12 and $x^3y^4z^5=(0.1)(600)^3$. Then $x^3+y^3+z^3$ is equal to :

The value of $\displaystyle\sum_{r=1}^{15}r^2\left(\frac{{}^{15}C_r}{{}^{15}C_{r-1}}\right)$ is equal to :

Let z = 1 + ai be a complex number, a > 0, such that $z^3$ is a real number. Then the sum $1+z+z^2+\dots+z^{11}$ is equal to :

Let $a_1, a_2, a_3, \dots, a_n, \dots$ be in A.P. If $a_3+a_7+a_{11}+a_{15}=72$, then the sum of its first 17 terms is equal to :

The sum $\displaystyle\sum_{r=1}^{10}(r^2+1)\times(r!)$ is equal to :

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