# Problems involving Geometric Progressions - Normal Problems with Solutions

Problem №1
Given a geometric progression ${a_n}$, for which $a_1=15$ and $q=-4$ find its sixth member.
Problem №2
Find the second member of a geometric progression $\{a_n\}$, which satisfies
$\begin{array}{|l}a_2+a_5-a_4=10\\a_3+a_6-a_5=20\end{array}$
Problem №3
Find $0,272727(27)$ as a fraction.
Problem №4
The sum of the members of an infinite geometric progression is $S_1=6$. The sum of the squares of all members of same progression is $S_2=18$. Find the first member of the progression.
Problem №5
Deterine the quotient q of a geometric progression $\{a_n\}$, for which $a_1=1$ and $S_4=40$
Problem №6
Find the quotient q of an infinite geometric progression $\{a_n\}$, for which S=15 and a_1=9
Problem №7
Find the quotient q of an infinite geometric progression $\{a_n\}$, for which $S=7$ and $a_1=4$
Problem №8
Find the sum of the first four members of the geometric progression ${a_n}$, for which $a_n=\frac{2.3^n}{5}$
Problem №9
Find the sum of the infinite geometric progression, explicitly defined by $a_n=\frac{2^n}{3^{n+1}}$
Problem №10
Find the sum of the infinite geometric progression $a_n=6.(\frac{1}{3})^n$
Problem №11
Find the product of the first 7 members of the geometric progression ${a_n}$, defined as:
$a_1=\frac{2}{11^3}$, $q=11$.
Problem №12
Let ${a_n}$ be an alternating geometric progression. If $a_1=5$ and $a_7=405$, determine the value of $a_4$
Problem №13
Find the sum of the first 5 powers of 7.
Problem №14
Let $a_n$ be a geometric progression, defined as $a_1=2$ and $q=-2$. Find the sum of its\' first 10 elements.
Problem №15
Find the first term of a geometric progression that second term is 2 and sum to infinity is 8.
Problem №16
Let $x_1, x_2$ be the roots of the equation $x^2-3x+a=0$ and $y_1,y_2$ be the roots of the equation $x^2-12x-b=0$. If $x_1,x_2,y_1,y_2$ form an increasing geometric progression in said order, determine the value of $a.b$.