Geometric Progressions

The sum of the first $10$ terms of the series $\sqrt{2}+\sqrt{4}+\sqrt{8}+............$ is equal to_____.

If $1 + 10 + {10}^2 + .............$ upto $n$ terms$=\frac{{10}^n-1}{9}$, then the sum of the series $4 + 44 + 444 + .........$ upto $n$ term is _____.

How many terms are there in the G.P. $3, 6, 12, 24,......,384$?

What should come in place of $?$ in the following number series?

How many terms are there in the G.P.$3, 6, 12, 24, ... , 384 ?$

In the series $7, 14, 28, ...$, what will be the ${10}^{th}$ term ?

What should come in place of $?$ in the following number series?

Let $f\left(x\right)=\tan x-{\tan }^{3}x+{\tan }^{5}x-{\tan }^{7}x+\dots \dots \dots ,\infty$, where $x\in \left(0,\frac{\pi }{4}\right)$, then which of the follow is/are correct?

If $1, 3, 3^2, 3^3, \dots .....,3^{2021}$ is divided by $50$, then remainder is

Determine whether the infinite geometric series below are converging, and also, find the sum for the following series : $0.25+0.375+0.5625+\dots$

Let $x_1,x_2,\dots ,x_6$, be the roots of the polynomial equation $x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0$. Then

Find the value of $k$ such that $3k-1 , 5k , 11k-2$ are in geometric progression and $k>1$ :

Find the sixth term for the sequence $1 , 2.50 , 6.25 , \dots \dots \dots \dots$ :

What will be the sum of first $10$ terms of the sequence $20, -10, 5, -2.5\dots \dots \dots \dots \dots$ ?

Let $a,b$ be distinct positive real numbers, whose geometric mean equals $\frac{a^{t-99}+b^{t-99}}{a^{t-100}+b^{t-100}}$. Then $t$ must equal

Let $A_1,{ A}_2,{ A}_3,\dots \dots .$ be an increasing geometric progression. If $\frac{1}{{ A}_4}+\frac{1}{{ A}_8}=\frac{10}{63}$ and $A_5{ A}_7=441$, then $3\left({ A}_2+A_4\right)$ is equal to ___________.

The sum of infinite number of terms of a Geometric Progression is $3$ and the sum of the squares of its terms is $\frac{9}{2}$. The sum of the cubes of its terms is

Let $\left|a\right|<1, \left|b\right|<1, \left|c\right|<1$ and $x=\sum _{n=0}^{\infty }{a}^n, y=\sum _{n=0}^{\infty }{b}^n, z=\sum _{n=0}^{\infty }{c}^n$. If $\frac{1}{x}, \frac{1}{y} \text{and} \frac{1}{z}$ are in Arithmetic Progression, then

Let $a_1, a_2, a_3, a_4, a_5$ be a $\mathrm{GP}$ of positive real number such that the $\mathrm{AM}$ of $a_2$ and $a_4$ is $117$ and the $\mathrm{GM}$ of $a_2$ and $a_4$ is $108$. Then, the $\mathrm{AM}$ of $a_1$ and $a_5$ is