Geometric progression

Let for $n=1,2,\dots,50$, $S_n$ be the sum of the infinite geometric progression whose first term is $n^2$ and whose common ratio is $\dfrac1{(n+1)^2}$. Then the value of $\dfrac1{26}+\displaystyle\sum_{n=1}^{50}\left(S_n+\frac2{n+1}-n-1\right)$ is equal to ______.

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\dots+A^{10}$ is equal to :

Let $M=\begin{bmatrix}0&-\alpha\\\alpha&0\end{bmatrix}$, where $\alpha$ is a non-zero real number an $N=\displaystyle\sum_{k=1}^{49}M^{2k}$. If $\left(I-M^2\right)N=-2I$, then the positive integral value of $\alpha$ is ______.

Consider two G.Ps. 2, $2^2$, $2^3$, ... and 4, $4^2$, $4^3$, ... of 60 and n terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}8}$, then $\displaystyle\sum_{k=1}^nk(n-k)$ is equal to :

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5. Let the sum of its first five terms be $\dfrac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10ar$, $n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$, is equal to :

Let $x_1,x_2,x_3,\ldots,x_{20}$ be in geometric progression with $x_1=3$ and the common ratio $\dfrac12$. A new data is constructed replacing each $x_i$ by $(x_i-i)^2$. If $\bar x$ is the mean of new data, then the greatest integer less than or equal to $\bar x$ is ______.

If $\dfrac6{3^{12}}+\dfrac{10}{3^{11}}+\dfrac{20}{3^{10}}+\dfrac{40}{3^9}+\ldots+\dfrac{10240}3=2^n\cdot m$, where m is odd, then m · n is equal to ______.

The coefficient of $x^{1011}$ in the expression $(1+2x)^{2022}+2x(1+2x)^{2021}+(2x)^2(1+2x)^{2020}+\dots+(2x)^{2022}$ is

Let $A_k=\displaystyle\int_{k\pi}^{(k+1)\pi}e^{-x}\sin x\,dx$, k = 0, 1, 2, ..., 20. Then $\displaystyle\sum_{k=0}^{20}|A_k|$ is equal to

Consider $M$ with $r = \dfrac{1025}{513}$. Let $k$ be the number of all those circles $C_n$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

Consider $M$ with $r = \dfrac{(2^{199} - 1)\sqrt2}{2^{198}}$. The number of all those circles $D_n$ that are inside $M$ is

If $e^{\left(\cos^2x+\cos^4x+\cos^6x+\dots\infty\right)\log_e2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\dfrac{2\sin x}{\sin x+\sqrt3\cos x}\left(0<x<\dfrac\pi2\right)$ is :

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_1 = b_1 = c$, then the number of all possible values of $c$, for which the equality

$$2(a_1 + a_2 + \cdots + a_n) = b_1 + b_2 + \cdots + b_n$$

holds for some positive integer $n$, is ______

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:

$E_1: \dfrac{x^2}9 + \dfrac{y^2}4 = 1$;

$R_1$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;

$E_n$: ellipse $\dfrac{x^2}{a_n^2} + \dfrac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}$, $n > 1$;

$R_n$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_n$, $n > 1$.

Then which of the following options is/are correct?

The number of real solutions of the equation

$$\sin^{-1}\left(\sum_{i=1}^{\infty} x^{i+1} - x \sum_{i=1}^{\infty} \left(\frac{x}{2}\right)^i\right) = \frac{\pi}{2} - \cos^{-1}\left(\sum_{i=1}^{\infty} \left(-\frac{x}{2}\right)^i - \sum_{i=1}^{\infty} (-x)^i\right)$$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is _____ .

(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\cos^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

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

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 :

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 :

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ($l,n>1$) and $G_1$, $G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2G_2^4+G_3^4$ equals.

The sum of the 3$^{\text{rd}}$ and the 4$^{\text{th}}$ terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7$^{\text{th}}$ term is :