Geometric progression

Let $a_1,a_2,a_3,\dots$ be an A.P. and $g_1=a_1,g_2,g_3,\dots$ be an increasing G.P. If $a_1=a_2+g_2=1$ and $a_3+g_3=4$, then $a_{10}+g_5$ is equal to :

Let $\alpha=\dfrac14+\dfrac18+\dfrac1{16}+\dots\infty$ and $\beta=\dfrac13+\dfrac19+\dfrac1{27}+\dots\infty$. Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

Let $\alpha,\beta$ be the roots of the equation $x^2-x+p=0$ and $\gamma,\delta$ be the roots the equation $x^2-4x+q=0$; $p,q\in\mathbb Z$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then $|p+q|$ equals :

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 :

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 $7\overbrace{5\cdots5}^{r}7$ 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
$$\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac\pi6\right).$$
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?

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 :

If $\dfrac1{2\cdot3^{10}}+\dfrac1{2^2\cdot3^9}+\dots+\dfrac1{2^{10}\cdot3}=\dfrac K{2^{10}\cdot3^{10}}$, then the remainder when K is divided by 6 is :

The greatest integer less than or equal to the sum of first 100 terms of the sequence $\dfrac13,\dfrac59,\dfrac{19}{27},\dfrac{65}{81},\dots$ is equal to ______.

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\dots+x^{500}$, $x>0$, is

If $A=\displaystyle\sum_{n=1}^\infty\frac1{\left(3+(-1)^n\right)^n}$ and $B=\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\dfrac AB$ is equal to :

If $a_1\,(>0)$, $a_2$, $a_3$, $a_4$, $a_5$ are in a G.P., $a_2+a_4=2a_3+1$ and $3a_2+a_3=2a_4$, then $a_2+a_4+2a_5$ is equal to ______.

If $x=\displaystyle\sum_{n=0}^\infty a^n$, $y=\displaystyle\sum_{n=0}^\infty b^n$, $z=\displaystyle\sum_{n=0}^\infty c^n$, where $a,b,c$ are in A.P. and $|a|<1$, $|b|<1$, $|c|<1$, $abc\ne0$, then :

Let $A_1,A_2,A_3,\dots$ be an increasing geometric progression of positive real numbers. If $A_1A_3A_5A_7=\dfrac1{1296}$ and $A_2+A_4=\dfrac7{36}$, then, the value of $A_6+A_8+A_{10}$ is equal to