Sequences and Series

If
$$\Delta_r=\begin{vmatrix}r&2r-1&3r-2\\\dfrac n2&n-1&a\\\dfrac12n(n-1)&(n-1)^2&\dfrac12(n-1)(3n+4)\end{vmatrix},$$
then the value of $\displaystyle\sum_{r=1}^{n-1}\Delta_r$ :

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac12$, then the number of terms in the A.P. is :

Let $f(n)=\left[\dfrac13+\dfrac{3n}{100}\right]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\displaystyle\sum_{n=1}^{56}f(n)$ is equal to :

The value of $\cot\left(\displaystyle\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^n2k\right)\right)$ is

Let $S_n=\displaystyle\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}2}k^2$. Then $S_n$ can take value(s)

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k-20=$

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$, with $\alpha>\beta$. If $a_n=\alpha^n-\beta^n$ for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is

Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$, then
$$\sum_{n=0}^\infty\left(\frac1\alpha+\frac1\beta\right)^n$$
is

Let $a_1,a_2,a_3,\ldots,a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\displaystyle\sum_{i=1}^pa_i$, $1\le p\le100$. For any integer $n$ with $1\le n\le20$, let $m=5n$. If $\dfrac{S_m}{S_n}$ does not depend on $n$, then $a_2$ is

The minimum value of the sum of real numbers $a^{-5}$, $a^{-4}$, $3a^{-3}$, $1$, $a^8$ and $a^{10}$ with $a>0$ is

If
$$\lim_{x\to0}\left[1+x\ln(1+b^2)\right]^{\frac1x}=2b\sin^2\theta,\quad b>0\ \text{and}\ \theta\in(-\pi,\pi],$$
then the value of $\theta$ is

Let $S_k$, $k=1,2,\ldots,100$, denote the sum of the infinite geometric series whose first term is $\dfrac{k-1}{k!}$ and the common ratio is $\dfrac1k$. Then the value of $\dfrac{100^2}{100!}+\displaystyle\sum_{k=1}^{100}\left|(k^2-3k+1)S_k\right|$ is

Let $a_1,a_2,a_3,\ldots,a_{11}$ be real numbers satisfying $$a_1=15,\quad 27-2a_2>0\ \text{ and }\ a_k=2a_{k-1}-a_{k-2}\ \text{ for } k=3,4,\ldots,11.$$ If $\dfrac{a_1^2+a_2^2+\cdots+a_{11}^2}{11}=90$, then the value of $\dfrac{a_1+a_2+\cdots+a_{11}}{11}$ is equal to

If
$$\frac{\sin^4x}2+\frac{\cos^4x}3=\frac15,$$
then

If the sum of first $n$ terms of an A.P. is $cn^2$, then the sum of squares of these $n$ terms is

A straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then

Let $g(x)=\log f(x)$ where $f(x)$ is a twice differentiable positive function on $(0,\infty)$ such that $f(x+1)=xf(x)$. Then, for $N=1,2,3,\ldots$,
$$g''\left(N+\frac12\right)-g''\left(\frac12\right)=$$

Suppose four distinct positive numbers $a_1,a_2,a_3,a_4$ are in G.P. Let $b_1=a_1$, $b_2=b_1+a_2$, $b_3=b_2+a_3$ and $b_4=b_3+a_4$.

STATEMENT-1: The numbers $b_1,b_2,b_3,b_4$ are neither in A.P. nor in G.P.

and

STATEMENT-2: The numbers $b_1,b_2,b_3,b_4$ are in H.P.

The sum $V_1+V_2+\cdots+V_n$ is

$T_r$ is always