Sequences and Series

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

For the functions $f(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta$, and $g(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta$, $\alpha>\beta>0$, let $\displaystyle\min_{0<\theta<\frac\pi2}f(\theta)=\max_{0<\theta<\pi}g(\theta)$. If the first term of a G.P. is $\left(\dfrac\alpha{2\beta}\right)$, its common ratio is $\left(\dfrac{2\beta}\alpha\right)$ and the sum of its first 10 terms is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

Consider the quadratic equation $(n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0$, $n\in\mathbf R$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\dfrac\alpha\beta$, is :

The sum $1+\dfrac12\left(1^2+2^2\right)+\dfrac13\left(1^2+2^2+3^2\right)+\dots$ upto 10 terms is equal to :

Let $\alpha=3+4+8+9+13+14+\dots$ upto 40 terms. If $(\tan\beta)^{\frac\alpha{1020}}$ is a root of the equation $x^2+x-2=0$, $\beta\in\left(0,\dfrac\pi2\right)$, then $\sin^2\beta+3\cos^2\beta$ is equal to :

Let $f$ be a polynomial function such that
$$\log_2(f(x))=\left(\log_2\left(2+\frac23+\frac29+\dots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right),$$
$x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to ________.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) > 0$ for all $x \in \mathbb{R}$, and $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$.

Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31}) = 64f(a_{25})$, and

$$\sum_{i=1}^{50} f(a_i) = 3\left(2^{25} + 1\right),$$

then the value of

$$\sum_{i=6}^{30} f(a_i)$$

is _____.

Let

$$\alpha = \frac{1}{\sin60^\circ\sin61^\circ} + \frac{1}{\sin62^\circ\sin63^\circ} + \cdots + \frac{1}{\sin118^\circ\sin119^\circ}.$$

Then the value of

$$\left(\frac{\operatorname{cosec}1^\circ}{\alpha}\right)^2$$

is _____.

Let $7\overbrace{5\cdots5}^{r}7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7\overbrace{5\cdots5}^{98}7$. If $S = \dfrac{7\overbrace{5\cdots5}^{99}7 + m}{n}$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is _____.

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\dfrac13$, then the probability that the experiment stops with head is

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1d_2 = 10$. For each $i = 1, 2, \ldots, 100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$. If $A_{51} - A_{50} = 1000$, then the value of $A_{100} - A_{90}$ is _____.

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1 = 7$ and common difference 8. Let $T_1, T_2, T_3, \ldots$ be such that $T_1 = 3$ and $T_{n+1} - T_n = a_n$ for $n \ge 1$. Then, which of the following is/are TRUE?

Let
$$\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac\pi6\right).$$
Let $g : [0, 1] \to \mathbb{R}$ be the function defined by
$$g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}.$$
Then, which of the following statements is/are TRUE?

Let $S=\{\sqrt n:1\le n\le50$ and $n$ is odd$\}$.

Let $a\in S$ and $A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}$.

If $\displaystyle\sum_{a\in S}\det(\operatorname{adj}A)=100\lambda$, then $\lambda$ is equal to :

If $\{a_i\}_{i=1}^n$, where $n$ is an even integer, is an arithmetic progression with common difference 1, and $\displaystyle\sum_{i=1}^na_i=192$, $\displaystyle\sum_{i=1}^{n/2}a_{2i}=120$, then $n$ is equal to :

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

Let $x,y>0$. If $x^3y^2=2^{15}$, then the least value of $3x+2y$ is

The sum of all the elements of the set $\{\alpha\in\{1,2,\dots,100\}:\mathrm{HCF}(\alpha,24)=1\}$ is ______.

The remainder on dividing $1+3+3^2+3^3+\dots+3^{2021}$ by 50 is ______.

Let $f:\mathbb N\to\mathbb R$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which
$$\sum_{k=1}^{10}f(\alpha+k)=\frac{512}3\left(2^{20}-1\right)$$
holds, is :