Introduction to Geometric Progression

In the series 2, 6, 18, 54, ... what will be the 8th term ?

$2 +4+8+\dots \dots$. How many terms of the series to make the sum equal to $1022$.

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

Let the first term a and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to

If $a^2+{\left(ar\right)}^2+{\left(a{r}^2\right)}^2=33033,\left(a,r\in N\right)$ then the value of $a+ar+a{r}^2$ is

If $a_1,a_2,.....,a_n$ be in $A.P.$ and $a_3,a_5,a_8,b_1,b_2,.....b_n$ be in $G.P.$ If $a_9=40,$ then the value of $\sum _{i=1}^9{a_i}^2$ is 

Three numbers is $\mathrm{G}.\mathrm{P}.$ with their sum is $130$ and their product is $27,000$ are _________

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-4x+\lambda =0$ and let $\gamma$ and $\delta$ be the roots of the equation $x^2-64x+\mu =0.$ If $\alpha , \beta , \gamma , \delta$ are in geometric progression with positive common ratio, then the value of $\lambda$ is

Direction Each of the following questions is followed by two statements:

What is the middle number of $7$consecutive whole numbers?

$\left(\mathrm{I}\right)$ Product of number is $702,800$ .

$\left(\mathrm{II}\right)$ Sum of the number is $105$ .

Given $\mathrm{A}=2^{65}$ and $\mathrm{B}=\left(2^{64}+2^{63}+2^{62}+\dots +2°\right)$, which of the following is true?

The numbers $x, 8, y$ are in $\mathrm{G}.\mathrm{P}.$and the numbers $x, y,-8$ are in $\mathrm{A}.\mathrm{P}.$The values of $x, y$ are _______.

If the continued product of three numbers in $\mathrm{G}.\mathrm{P}.$ is $27$and the sum of their products in pairs is $39$the numbers are_____________

Find five numbers in $\mathrm{G}.\mathrm{P}.$ such that their product is $32$and the product of the last two is $108$.

Three numbers in $\mathrm{G}.\mathrm{P}$. with their sum $\frac{13}{3}$and sum of their squares $\frac{91}{9}$ are

Three numbers in $\mathrm{G}.\mathrm{P}$. with their sum $130$and their product $27,000$are__________

If the first term of a $\mathrm{G}.\mathrm{P}.$ exceeds the second term by $2$and the sum to infinity is $50$the series is____________.

The sum upto infinity of the series $\frac{2}{3}+\frac{5}{9}+\frac{2}{27}+\frac{5}{81}+\dots \dots .$ is

The number of subsets of the set $\{2,3,5\}$ is

If $x, y, z$ are in $\mathrm{G}.\mathrm{P}.$, then _____.

$t_4$ of a $\mathrm{G}.\mathrm{P}.$ in $x,t_{10}=y$ and $t_{16}=z$. Then _____.